Calculus

Problem 33501

Find the derivative $y'$ , where $y=(\ln x)^{\ln x}$ .

See Solution

Problem 33502

Find the derivative of $y=x \sin \left(\log _{7} x\right)$ . What is $y^{\prime}=$ ?

See Solution

Problem 33503

Find the derivative $y'$ , given $y=3 \log _{8}(\log _{2} t)$ .

See Solution

Problem 33504

Evaluate the integral from $e^{2}$ to $e^{3}$ of $\frac{1+\ln (x)}{x \ln (x)} \, dx$ . Options: $1-\ln(2)$ , $1+\ln(3)-\ln(2)$ , $\ln(3)-\ln(2)$ , $1-2\ln(2)$ , none.

See Solution

Problem 33505

Find the derivative of $y=x \ln (\ln (x))$ at $x=e$ . What is the value? Choose from: $e$ , none, 2, 0, 1.

See Solution

Problem 33506

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value? Options: none, e, 1, 0, $e^{-1}$ .

See Solution

Problem 33507

Find $y^{\prime}$ if $y=3 \log _{8}(\log _{2} t)$ . Choose the correct derivative from the options.

See Solution

Problem 33508

Find the derivative $y'$ , where $y=(x+1)^{x}$ . Options include: $(x+1)^{x}\left(\frac{x}{x+1}+\ln (x+1)\right)$ .

See Solution

Problem 33509

Find the derivative $y'$ , where $y=3 \log _{8}(\log _{2} t)$ .

See Solution

Problem 33510

Find the derivative of $y=x^{\ln (x)}$ for $x>0$ . Choose the correct option from the list.

See Solution

Problem 33511

Evaluate the integral $\int \frac{d x}{x \log_{10} x}$ . Select the correct answer from the options given.

See Solution

Problem 33512

Find the derivative of $y=(x+1)^{x}$ . What is $y^{\prime}=$ ?

See Solution

Problem 33513

Find the derivative $y^{\prime}$ of $y=3 \log _{8}\left(\log _{2} t\right)$ .

See Solution

Problem 33514

Find the derivative $y^{\prime}$ if $y=\log _{2}(8 t^{\ln 2})$ . Choices: $\frac{\ln 8}{t}$ , $3+\ln t$ , $\frac{t}{\ln 2}$ , $\frac{1}{t}$ , none.

See Solution

Problem 33515

Calculate the integral from $e^{2}$ to $e^{3}$ of $\frac{1+\ln (x)}{x \ln (x)}$ . What is the result?

See Solution

Problem 33516

Find the derivative $y'$ , where $y=3 \log _{8}(\log _{2} t)$ .

See Solution

Problem 33517

Find the derivative $y'$ , where $y=(x+1)^{x}$ . Options include: $\frac{x}{x+1}+\ln (x+1)$ , $(x+1)^{x} \frac{x}{x+1}$ , none, $(x+1)^{x}\left(\frac{1}{x+1}+\ln (x+1)\right)$ , $(x+1)^{x}\left(\frac{x}{x+1}+\ln (x+1)\right)$ .

See Solution

Problem 33518

Find the derivative $y'$ , where $y=x \sin \left(\log _{7} x\right)$ . Choose the correct option from the list.

See Solution

Problem 33519

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x))$ . What is the result? Options: 5, none, 23, 1, 21.

See Solution

Problem 33520

Evaluate the integral $\int \frac{2 \log _{2}(x-1)}{(x-1)} d x$ and choose the correct answer.

See Solution

Problem 33521

Evaluate the integral from 2 to e of $\frac{1+\ln (x)}{x \ln (x)} \, dx$ . Select the correct answer.

See Solution

Problem 33522

Evaluate the integral from 2 to e of $\frac{1+\ln (x)}{x \ln (x)}$ dx. What is the result?

See Solution

Problem 33523

Find $y'$ if $y=(x+1)^{x}$ . Choose from:
1. $(x+1)^{x}\left(\frac{1}{x+1}+\ln (x+1)\right)$
2. $(x+1)^{x}\left(\frac{x}{x+1}+\ln (x+1)\right)$
3. none
4. $\frac{x}{x+1}+\ln (x+1)$
5. $(x+1)^{x} \frac{x}{x+1}$

See Solution

Problem 33524

Find the derivative of $y=x^{\ln (x)}$ for $x>0$ . Choose the correct option from the list provided.

See Solution

Problem 33525

Find the limits of the piecewise function $f(x)=\left\{\begin{array}{lll}-\frac{3}{x+4} & \text { if } & x<-4 \\ 4 x+17 & \text { if } & x>-4\end{array}\right.$ at $x=-4$ .

See Solution

Problem 33526

Find the derivative $y^{\prime}$ of $y=(x+1)^{x}$ . Choose from the options given.

See Solution

Problem 33527

Given $0 \leq a_{n} \leq b_{n}$ for $n \geq 1$ , which statements about the convergence/divergence of $\sum a_{n}$ and $\sum b_{n}$ are true?

See Solution

Problem 33528

Find the derivative of $y=x \ln (\ln (x))$ at $x=e$ . What is the value? Options: 2, e, none, 0, 1.

See Solution

Problem 33529

Integrate $x^{2 x}(1+\ln x)$ with respect to $x$ . Choose the correct answer.

See Solution

Problem 33530

Evaluate the integral: $\int \frac{d x}{x \log _{10} x}$ . Choose the correct answer from the options provided.

See Solution

Problem 33531

Find the derivative $y'$ , if $y=(\sqrt{t})^{t}$ .

See Solution

Problem 33532

Find $y'$ if $y=(\sqrt{t})^{t}$ . Choose one: $\frac{1}{2 \sqrt{t}}$ , none, $\frac{\ln t}{2}+\frac{1}{2}$ , $(\sqrt{t})^{t}\left(\frac{\ln t}{2}+\frac{1}{2}\right)$ , $t(\sqrt{t})^{t-1}$ .

See Solution

Problem 33533

Determine the convergence of these series: 1. $\sum_{n=1}^{\infty} \frac{2^{n}+n^{4}}{4^{n}+n^{2}}$ , 2. $\sum_{n=2}^{\infty} \frac{4^{n}}{5^{n}+n}$ , 3. $\sum_{n=1}^{\infty} \frac{n !}{3^{n}}$ .

See Solution

Problem 33534

Determine which series converge:
1. $\sum_{n=1}^{\infty} \frac{\ln n}{e^{n}+2}$
2. $\sum_{n=2}^{\infty} \frac{\sin ^{2}(n)}{n+n^{3 / 2}}$
3. $\sum_{n=1}^{\infty} \frac{1}{n^{7 / 3}}$ Options: a. 2 and 3, b. 1 only, c. all, d. 2 only, e. 1 and 2, f. none, g. 1 and 3, h. 3 only.

See Solution

Problem 33535

Find the derivative of $y=(\ln x)^{\ln x}$ . What is $y^{\prime}=$ ?

See Solution

Problem 33536

If $y = 3 \log_{8}(\log_{2} t)$ , find the derivative $y'$ .

See Solution

Problem 33537

Evaluate the integral $\int x^{2 x}(1+\ln x) d x$ .

See Solution

Problem 33538

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x))$ . What is the value? Options: 5, none, 23, 1, 21.

See Solution

Problem 33539

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$

See Solution

Problem 33540

Calculate the integral from 2 to 3 of $x^x(1+\ln(x))$ . What is the result? Options: 5, 21, none, 1, 23.

See Solution

Problem 33541

Differentiate the function $f(x) = x^{\ln(x)}$ with respect to $x$ .

See Solution

Problem 33542

Determine if the series $\sum_{n=1}^{\infty} \frac{7+10 \cos n}{n^{4}}$ converges or diverges.

See Solution

Problem 33543

Evaluate the integral from 1 to 2: $\int_{1}^{2} x^{x}(1+\ln (x)) d x=$ . Choose from: 0, 1, 2, 3, none.

See Solution

Problem 33544

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value?

See Solution

Problem 33545

Find the derivative of $y=x^{\ln (x)}$ for $x>0$ . Select one:

See Solution

Problem 33546

Find the derivative $y^{\prime}$ if $y=(\ln x)^{\ln x}$ .

See Solution

Problem 33547

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct answer.

See Solution

Problem 33548

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct answer.

See Solution

Problem 33549

Calculate the integral from 1 to 2 of the function $x^{x}(1+\ln (x))$ .

See Solution

Problem 33550

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct answer from the options given.

See Solution

Problem 33551

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{x^{2}+5 x-6}{-2 x+2}=$

See Solution

Problem 33552

Evaluate the integral: $\int \frac{d x}{x \log_{10} x}$ . Choose the correct answer from the options given.

See Solution

Problem 33553

Find the derivative of $y=\sin \left(x^{x}\right)$ . What is $y^{\prime}=$ ?

See Solution

Problem 33554

Given $0 \leq a_{n} \leq b_{n}$ , determine which statement about the convergence of $\sum a_{n}$ and $\sum b_{n}$ is true.

See Solution

Problem 33555

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value? Options: $e$ , 0, 1, $e^{-1}$ , none.

See Solution

Problem 33556

Find the derivative of the function $y=x^{\sin x}$ . What is $y^{\prime}$ ?

See Solution

Problem 33557

Given $0 \leq a_{n} \leq b_{n}$ , determine which statement is true about the series: a, b, c, or d.

See Solution

Problem 33558

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct answer from the options provided.

See Solution

Problem 33559

Find the derivative of $y=x \ln (\ln (x))$ at $x=e$ . What is the value? Options: 2, e, 0, 1, none.

See Solution

Problem 33560

Find the derivative of the function $y=x \sin \left(\log _{7} x\right)$ . What is $y^{\prime}$ ?

See Solution

Problem 33561

Find the derivative $y'$ if $y=t \log _{3}(e^{(\sin t)(\ln 3)})$ .

See Solution

Problem 33562

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{(\ln n)^{n}}{n^{n}}$ using various tests.

See Solution

Problem 33563

Determine the convergence of these series: I. $\sum_{n=1}^{\infty} \frac{1}{1+\frac{1}{n}}$ , II. $\sum_{n=2}^{\infty} \frac{1}{\left(\ln n^{4}\right)}$ , III. $\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^{2}+1}}$ .

See Solution

Problem 33564

Determine if the series $\sum_{n=1}^{\infty} \frac{1}{n 3^{n}}$ converges or diverges using comparison tests.

See Solution

Problem 33565

Evaluate the integral from 2 to e of $\frac{1+\ln (x)}{x \ln (x)} dx$ . Choose the correct answer from the options.

See Solution

Problem 33566

Find the derivative $y^{\prime}$ of the function $y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)$ .

See Solution

Problem 33567

Find $y^{\prime}$ if $y=(\sqrt{t})^{t}$ . Choose the correct option from: 1) $\frac{1}{2 \sqrt{t}}$ , 2) none, 3) $\frac{\ln t}{2}+\frac{1}{2}$ , 4) $(\sqrt{t})^{t}\left(\frac{\ln t}{2}+\frac{1}{2}\right)$ , 5) $t(\sqrt{t})^{t-1}$ .

See Solution

Problem 33568

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{5 n}{n^{2}+1}$ using various tests.

See Solution

Problem 33569

Determine if the series $\sum_{n=1}^{\infty} \frac{n^{n}}{5^{n^{2}}}$ converges or diverges.

See Solution

Problem 33570

Find the derivative of $y=x^{\sin x}$ . What is $y^{\prime}=$ ? Choose from the given options.

See Solution

Problem 33571

Find the limit $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}$ for the series $\sum_{n=1}^{\infty} \frac{(2 n) !}{(n !)(n+1) !}$ . Options: a. 2 b. 4 c. 0 d. 1 e. $\infty$

See Solution

Problem 33572

Find the derivative $y^{\prime}$ of $y=\log_{2}(8 t^{\ln 2})$ . Select from options given.

See Solution

Problem 33573

Evaluate the integral: $\int \frac{2 \log _{2}(x-1)}{(x-1)} d x$ . Choose the correct answer from the options.

See Solution

Problem 33574

Find the derivative of $y=\ln (\sec \theta)+\ln \left(\cos ^{2} \theta\right)$ . What is it?

See Solution

Problem 33575

Find the derivative $y^{\prime}$ of the function $y=x^{\sin x}$ .

See Solution

Problem 33576

Find the derivative $y^{\prime}$ if $y=3 \log _{8}\left(\log _{2} t\right)$ .

See Solution

Problem 33577

Find the derivative $y'$ , given $y=3 \log _{8}(\log _{2} t)$ .

See Solution

Problem 33578

Determine the convergence of the series $\sum_{n=2}^{\infty} \frac{1}{\sqrt{n^{2}-1}}$ using comparison tests.

See Solution

Problem 33579

Evaluate the integral from 2 to e of $\frac{1+\ln (x)}{x \ln (x)} dx$ and choose the correct answer.

See Solution

Problem 33580

Find the limit $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}$ for the series $\sum_{n=1}^{\infty} \frac{3^{1-2 n}}{n^{2}+1}$ . Choices: a. $\infty$ , b. $\frac{1}{3}$ , c. 9, d. 3, e. $\frac{1}{9}$ .

See Solution

Problem 33581

Determine if the series $\sum_{n=1}^{\infty} \frac{5 n}{n^{2}+1}$ converges or diverges using comparison tests.

See Solution

Problem 33582

Find the integral of $\frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct answer.

See Solution

Problem 33583

Find the derivative of $y=\sin \left(x^{x}\right)$ : what is $y^{\prime}$ ? Choose from the options.

See Solution

Problem 33584

Find the derivative $y^{\prime}$ of $y=x \sin \left(\log _{7} x\right)$ .

See Solution

Problem 33585

Find the derivative of $y=x^{\sin x}$ , which is $y^{\prime}=$ . Choose the correct option from the list provided.

See Solution

Problem 33586

Find the derivative $y'$ , where $y=(x+1)^{x}$ . Choose the correct expression for $y'$ .

See Solution

Problem 33587

Calculate the integral: $\int_{1}^{e^{3}} \frac{1+\ln (x)}{x \ln (x)} d x$

See Solution

Problem 33588

Evaluate the integral $\int_{1}^{2} x^{x}(1+\ln (x)) d x=$ : Select one 0, 20, none, 3, 10.

See Solution

Problem 33589

Determine which series converge:
1. $\sum_{n=1}^{\infty} \frac{\ln n}{e^{n}+2}$
2. $\sum_{n=2}^{\infty} \frac{\sin^2(n)}{n+n^{3/2}}$
3. $\sum_{n=1}^{\infty} \frac{1}{n^{7/3}}$ Options: a. I and III, b. I only, c. None, d. all, e. III only, f. III and II, g. I and II, h. II only.

See Solution

Problem 33590

Find the derivative $y^{\prime}$ of $y=(x+1)^{x}$ . Options include:
1. $\frac{x}{x+1}+\ln (x+1)$
2. $(x+1)^{x}\left(\frac{x}{x+1}+\ln (x+1)\right)$
3. none
4. $(x+1)^{x} \frac{x}{x+1}$
5. $(x+1)^{x}\left(\frac{1}{x+1}+\ln (x+1)\right)$

See Solution

Problem 33591

If $0 \leq a_{n} \leq b_{n}$ , which statements about the convergence of $\sum a_{n}$ and $\sum b_{n}$ are true?

See Solution

Problem 33592

Find the limit as $x$ approaches 8 for the expression $9$ .

See Solution

Problem 33593

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x))$ . What is the result? Options: 1, 23, 5, 21, none.

See Solution

Problem 33594

Find the derivative of $f(x)=9x+11$ and calculate $f'(-2)$ .

See Solution

Problem 33595

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{(\ln n)^{n}}{n^{n}}$ using various tests.

See Solution

Problem 33596

Evaluate the limit as $z$ approaches -3 for the expression $2 z^{5}-6 z$ .

See Solution

Problem 33597

Find the derivative $y'$ , if $y=\sin(x^x)$ . Choose from the options provided.

See Solution

Problem 33598

Find the derivative of $y=x^{\sin x}$ . What is $y^{\prime}=$ ?

See Solution

Problem 33599

Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{n+1}{n^{3}+6 n+3}$ using limit comparison tests.

See Solution

Problem 33600

Find the tangent line equation for $y=4 \sin x$ at $\left(\frac{\pi}{6}, 2\right)$ in the form $y=m x+b$ .

See Solution

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Calculus

Problem 33501

Find the derivative y′y'y′, where y=(ln⁡x)ln⁡xy=(\ln x)^{\ln x}y=(lnx)lnx.

Problem 33502

Find the derivative of y=xsin⁡(log⁡7x)y=x \sin \left(\log _{7} x\right)y=xsin(log7​x). What is y′=y^{\prime}=y′=?

Problem 33503

Find the derivative y′y'y′, given y=3log⁡8(log⁡2t)y=3 \log _{8}(\log _{2} t)y=3log8​(log2​t).

Problem 33504

Problem 33505

Find the derivative of y=xln⁡(ln⁡(x))y=x \ln (\ln (x))y=xln(ln(x)) at x=ex=ex=e. What is the value? Choose from: eee, none, 2, 0, 1.

Problem 33506

Find the derivative of y=xln⁡(ln⁡(x))y=x^{\ln (\ln (x))}y=xln(ln(x)) at x=ex=ex=e. What is the value? Options: none, e, 1, 0, e−1e^{-1}e−1.

Problem 33507

Find y′y^{\prime}y′ if y=3log⁡8(log⁡2t)y=3 \log _{8}(\log _{2} t)y=3log8​(log2​t). Choose the correct derivative from the options.

Problem 33508

Find the derivative y′y'y′, where y=(x+1)xy=(x+1)^{x}y=(x+1)x. Options include: (x+1)x(xx+1+ln⁡(x+1))(x+1)^{x}\left(\frac{x}{x+1}+\ln (x+1)\right)(x+1)x(x+1x​+ln(x+1)).

Problem 33509

Find the derivative y′y'y′, where y=3log⁡8(log⁡2t)y=3 \log _{8}(\log _{2} t)y=3log8​(log2​t).

Problem 33510

Find the derivative of y=xln⁡(x)y=x^{\ln (x)}y=xln(x) for x>0x>0x>0. Choose the correct option from the list.

Problem 33511

Evaluate the integral ∫dxxlog⁡10x\int \frac{d x}{x \log_{10} x}∫xlog10​xdx​. Select the correct answer from the options given.

Problem 33512

Find the derivative of y=(x+1)xy=(x+1)^{x}y=(x+1)x. What is y′=y^{\prime}=y′=?

Problem 33513

Find the derivative y′y^{\prime}y′ of y=3log⁡8(log⁡2t)y=3 \log _{8}\left(\log _{2} t\right)y=3log8​(log2​t).

Problem 33514

Find the derivative y′y^{\prime}y′ if y=log⁡2(8tln⁡2)y=\log _{2}(8 t^{\ln 2})y=log2​(8tln2). Choices: ln⁡8t\frac{\ln 8}{t}tln8​, 3+ln⁡t3+\ln t3+lnt, tln⁡2\frac{t}{\ln 2}ln2t​, 1t\frac{1}{t}t1​, none.

Problem 33515

Calculate the integral from e2e^{2}e2 to e3e^{3}e3 of 1+ln⁡(x)xln⁡(x)\frac{1+\ln (x)}{x \ln (x)}xln(x)1+ln(x)​. What is the result?

Problem 33516

Find the derivative y′y'y′, where y=3log⁡8(log⁡2t)y=3 \log _{8}(\log _{2} t)y=3log8​(log2​t).

Problem 33517

Problem 33518

Find the derivative y′y'y′, where y=xsin⁡(log⁡7x)y=x \sin \left(\log _{7} x\right)y=xsin(log7​x). Choose the correct option from the list.

Problem 33519

Calculate the integral from 2 to 3 of xx(1+ln⁡(x))x^{x}(1+\ln (x))xx(1+ln(x)). What is the result? Options: 5, none, 23, 1, 21.

Problem 33520

Evaluate the integral ∫2log⁡2(x−1)(x−1)dx\int \frac{2 \log _{2}(x-1)}{(x-1)} d x∫(x−1)2log2​(x−1)​dx and choose the correct answer.

Problem 33521

Evaluate the integral from 2 to e of 1+ln⁡(x)xln⁡(x) dx\frac{1+\ln (x)}{x \ln (x)} \, dxxln(x)1+ln(x)​dx. Select the correct answer.

Problem 33522

Evaluate the integral from 2 to e of 1+ln⁡(x)xln⁡(x)\frac{1+\ln (x)}{x \ln (x)}xln(x)1+ln(x)​ dx. What is the result?

Problem 33523

Problem 33524

Find the derivative of y=xln⁡(x)y=x^{\ln (x)}y=xln(x) for x>0x>0x>0. Choose the correct option from the list provided.

Problem 33525

Find the limits of the piecewise function f(x)={−3x+4 if x<−44x+17 if x>−4f(x)=\left\{\begin{array}{lll}-\frac{3}{x+4} & \text { if } & x<-4 \\ 4 x+17 & \text { if } & x>-4\end{array}\right.f(x)={−x+43​4x+17​ if if ​x<−4x>−4​ at x=−4x=-4x=−4.

Problem 33526

Find the derivative y′y^{\prime}y′ of y=(x+1)xy=(x+1)^{x}y=(x+1)x. Choose from the options given.

Problem 33527

Given 0≤an≤bn0 \leq a_{n} \leq b_{n}0≤an​≤bn​ for n≥1n \geq 1n≥1, which statements about the convergence/divergence of ∑an\sum a_{n}∑an​ and ∑bn\sum b_{n}∑bn​ are true?

Problem 33528

Find the derivative of y=xln⁡(ln⁡(x))y=x \ln (\ln (x))y=xln(ln(x)) at x=ex=ex=e. What is the value? Options: 2, e, none, 0, 1.

Problem 33529

Integrate x2x(1+ln⁡x)x^{2 x}(1+\ln x)x2x(1+lnx) with respect to xxx. Choose the correct answer.

Problem 33530

Evaluate the integral: ∫dxxlog⁡10x\int \frac{d x}{x \log _{10} x}∫xlog10​xdx​. Choose the correct answer from the options provided.

Problem 33531

Find the derivative y′y'y′, if y=(t)ty=(\sqrt{t})^{t}y=(t​)t.

Problem 33532

Find y′y'y′ if y=(t)ty=(\sqrt{t})^{t}y=(t​)t. Choose one: 12t\frac{1}{2 \sqrt{t}}2t​1​, none, ln⁡t2+12\frac{\ln t}{2}+\frac{1}{2}2lnt​+21​, (t)t(ln⁡t2+12)(\sqrt{t})^{t}\left(\frac{\ln t}{2}+\frac{1}{2}\right)(t​)t(2lnt​+21​), t(t)t−1t(\sqrt{t})^{t-1}t(t​)t−1.

Problem 33533

Determine the convergence of these series: 1. ∑n=1∞2n+n44n+n2\sum_{n=1}^{\infty} \frac{2^{n}+n^{4}}{4^{n}+n^{2}}∑n=1∞​4n+n22n+n4​, 2. ∑n=2∞4n5n+n\sum_{n=2}^{\infty} \frac{4^{n}}{5^{n}+n}∑n=2∞​5n+n4n​, 3. ∑n=1∞n!3n\sum_{n=1}^{\infty} \frac{n !}{3^{n}}∑n=1∞​3nn!​.

Problem 33534

Problem 33535

Find the derivative of y=(ln⁡x)ln⁡xy=(\ln x)^{\ln x}y=(lnx)lnx. What is y′=y^{\prime}=y′=?

Problem 33536

If y=3log⁡8(log⁡2t)y = 3 \log_{8}(\log_{2} t)y=3log8​(log2​t), find the derivative y′y'y′.

Problem 33537

Evaluate the integral ∫x2x(1+ln⁡x)dx\int x^{2 x}(1+\ln x) d x∫x2x(1+lnx)dx.

Problem 33538

Calculate the integral from 2 to 3 of xx(1+ln⁡(x))x^{x}(1+\ln (x))xx(1+ln(x)). What is the value? Options: 5, none, 23, 1, 21.

Problem 33539

Evaluate the integral: ∫dxx(log⁡8x)2\int \frac{d x}{x\left(\log _{8} x\right)^{2}}∫x(log8​x)2dx​

Problem 33540

Calculate the integral from 2 to 3 of xx(1+ln⁡(x))x^x(1+\ln(x))xx(1+ln(x)). What is the result? Options: 5, 21, none, 1, 23.

Problem 33541

Differentiate the function f(x)=xln⁡(x)f(x) = x^{\ln(x)}f(x)=xln(x) with respect to xxx.

Problem 33542

Find the derivative $y'$ , where $y=(\ln x)^{\ln x}$ .

Find the derivative of $y=x \sin \left(\log _{7} x\right)$ . What is $y^{\prime}=$ ?

Find the derivative $y'$ , given $y=3 \log _{8}(\log _{2} t)$ .

Find the derivative of $y=x \ln (\ln (x))$ at $x=e$ . What is the value? Choose from: $e$ , none, 2, 0, 1.

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value? Options: none, e, 1, 0, $e^{-1}$ .

Find $y^{\prime}$ if $y=3 \log _{8}(\log _{2} t)$ . Choose the correct derivative from the options.

Find the derivative $y'$ , where $y=(x+1)^{x}$ . Options include: $(x+1)^{x}\left(\frac{x}{x+1}+\ln (x+1)\right)$ .

Find the derivative $y'$ , where $y=3 \log _{8}(\log _{2} t)$ .

Find the derivative of $y=x^{\ln (x)}$ for $x>0$ . Choose the correct option from the list.

Evaluate the integral $\int \frac{d x}{x \log_{10} x}$ . Select the correct answer from the options given.

Find the derivative of $y=(x+1)^{x}$ . What is $y^{\prime}=$ ?

Find the derivative $y^{\prime}$ of $y=3 \log _{8}\left(\log _{2} t\right)$ .

Find the derivative $y^{\prime}$ if $y=\log _{2}(8 t^{\ln 2})$ . Choices: $\frac{\ln 8}{t}$ , $3+\ln t$ , $\frac{t}{\ln 2}$ , $\frac{1}{t}$ , none.

Calculate the integral from $e^{2}$ to $e^{3}$ of $\frac{1+\ln (x)}{x \ln (x)}$ . What is the result?

Find the derivative $y'$ , where $y=3 \log _{8}(\log _{2} t)$ .

Find the derivative $y'$ , where $y=x \sin \left(\log _{7} x\right)$ . Choose the correct option from the list.

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x))$ . What is the result? Options: 5, none, 23, 1, 21.

Evaluate the integral $\int \frac{2 \log _{2}(x-1)}{(x-1)} d x$ and choose the correct answer.

Evaluate the integral from 2 to e of $\frac{1+\ln (x)}{x \ln (x)} \, dx$ . Select the correct answer.

Evaluate the integral from 2 to e of $\frac{1+\ln (x)}{x \ln (x)}$ dx. What is the result?

Find the derivative of $y=x^{\ln (x)}$ for $x>0$ . Choose the correct option from the list provided.

Find the limits of the piecewise function $f(x)=\left\{\begin{array}{lll}-\frac{3}{x+4} & \text { if } & x<-4 \\ 4 x+17 & \text { if } & x>-4\end{array}\right.$ at $x=-4$ .

Find the derivative $y^{\prime}$ of $y=(x+1)^{x}$ . Choose from the options given.

Given $0 \leq a_{n} \leq b_{n}$ for $n \geq 1$ , which statements about the convergence/divergence of $\sum a_{n}$ and $\sum b_{n}$ are true?

Find the derivative of $y=x \ln (\ln (x))$ at $x=e$ . What is the value? Options: 2, e, none, 0, 1.

Integrate $x^{2 x}(1+\ln x)$ with respect to $x$ . Choose the correct answer.

Evaluate the integral: $\int \frac{d x}{x \log _{10} x}$ . Choose the correct answer from the options provided.

Find the derivative $y'$ , if $y=(\sqrt{t})^{t}$ .

Find $y'$ if $y=(\sqrt{t})^{t}$ . Choose one: $\frac{1}{2 \sqrt{t}}$ , none, $\frac{\ln t}{2}+\frac{1}{2}$ , $(\sqrt{t})^{t}\left(\frac{\ln t}{2}+\frac{1}{2}\right)$ , $t(\sqrt{t})^{t-1}$ .

Determine the convergence of these series: 1. $\sum_{n=1}^{\infty} \frac{2^{n}+n^{4}}{4^{n}+n^{2}}$ , 2. $\sum_{n=2}^{\infty} \frac{4^{n}}{5^{n}+n}$ , 3. $\sum_{n=1}^{\infty} \frac{n !}{3^{n}}$ .

Find the derivative of $y=(\ln x)^{\ln x}$ . What is $y^{\prime}=$ ?

If $y = 3 \log_{8}(\log_{2} t)$ , find the derivative $y'$ .

Evaluate the integral $\int x^{2 x}(1+\ln x) d x$ .

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x))$ . What is the value? Options: 5, none, 23, 1, 21.

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$

Calculate the integral from 2 to 3 of $x^x(1+\ln(x))$ . What is the result? Options: 5, 21, none, 1, 23.

Differentiate the function $f(x) = x^{\ln(x)}$ with respect to $x$ .

Determine if the series $\sum_{n=1}^{\infty} \frac{7+10 \cos n}{n^{4}}$ converges or diverges.

Evaluate the integral from 1 to 2: $\int_{1}^{2} x^{x}(1+\ln (x)) d x=$ . Choose from: 0, 1, 2, 3, none.

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value?

Find the derivative of $y=x^{\ln (x)}$ for $x>0$ . Select one:

Find the derivative $y^{\prime}$ if $y=(\ln x)^{\ln x}$ .

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct answer.

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct answer.

Calculate the integral from 1 to 2 of the function $x^{x}(1+\ln (x))$ .

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct answer from the options given.

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{x^{2}+5 x-6}{-2 x+2}=$

Evaluate the integral: $\int \frac{d x}{x \log_{10} x}$ . Choose the correct answer from the options given.

Find the derivative of $y=\sin \left(x^{x}\right)$ . What is $y^{\prime}=$ ?

Given $0 \leq a_{n} \leq b_{n}$ , determine which statement about the convergence of $\sum a_{n}$ and $\sum b_{n}$ is true.

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value? Options: $e$ , 0, 1, $e^{-1}$ , none.

Find the derivative of the function $y=x^{\sin x}$ . What is $y^{\prime}$ ?

Given $0 \leq a_{n} \leq b_{n}$ , determine which statement is true about the series: a, b, c, or d.

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct answer from the options provided.

Find the derivative of $y=x \ln (\ln (x))$ at $x=e$ . What is the value? Options: 2, e, 0, 1, none.

Find the derivative of the function $y=x \sin \left(\log _{7} x\right)$ . What is $y^{\prime}$ ?

Find the derivative $y'$ if $y=t \log _{3}(e^{(\sin t)(\ln 3)})$ .

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{(\ln n)^{n}}{n^{n}}$ using various tests.

Determine if the series $\sum_{n=1}^{\infty} \frac{1}{n 3^{n}}$ converges or diverges using comparison tests.

Evaluate the integral from 2 to e of $\frac{1+\ln (x)}{x \ln (x)} dx$ . Choose the correct answer from the options.

Find the derivative $y^{\prime}$ of the function $y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)$ .

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{5 n}{n^{2}+1}$ using various tests.

Determine if the series $\sum_{n=1}^{\infty} \frac{n^{n}}{5^{n^{2}}}$ converges or diverges.

Find the derivative of $y=x^{\sin x}$ . What is $y^{\prime}=$ ? Choose from the given options.

Find the limit $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}$ for the series $\sum_{n=1}^{\infty} \frac{(2 n) !}{(n !)(n+1) !}$ . Options: a. 2 b. 4 c. 0 d. 1 e. $\infty$

Find the derivative $y^{\prime}$ of $y=\log_{2}(8 t^{\ln 2})$ . Select from options given.

Evaluate the integral: $\int \frac{2 \log _{2}(x-1)}{(x-1)} d x$ . Choose the correct answer from the options.

Find the derivative of $y=\ln (\sec \theta)+\ln \left(\cos ^{2} \theta\right)$ . What is it?