Solved on Jan 23, 2024

Find the derivative of $y=\log_{8}(9-x-x^{5})$ .

STEP 1

Assumptions
1. We are given the function $y=\log _{8}\left(9-x-x^{5}\right)$ .
2. We need to find the derivative of $y$ with respect to $x$ , denoted as $y'$ .
3. We will use the chain rule and the logarithmic differentiation rule to find $y'$ .

STEP 2

The derivative of a logarithm with base $a$ is given by the formula:
$\frac{d}{dx} \log_a(u(x)) = \frac{u'(x)}{u(x) \ln(a)}$
where $u(x)$ is a differentiable function of $x$ .

STEP 3

Identify $u(x)$ in the given function:
$u(x) = 9 - x - x^5$

STEP 4

Find the derivative of $u(x)$ with respect to $x$ , denoted as $u'(x)$ :
$u'(x) = \frac{d}{dx}(9 - x - x^5)$

STEP 5

Differentiate each term of $u(x)$ separately:
$u'(x) = \frac{d}{dx}(9) - \frac{d}{dx}(x) - \frac{d}{dx}(x^5)$

STEP 6

Calculate the derivatives:
$u'(x) = 0 - 1 - 5x^4$

STEP 7

Simplify the expression for $u'(x)$ :
$u'(x) = -1 - 5x^4$

STEP 8

Now apply the logarithmic differentiation rule to find $y'$ :
$y' = \frac{u'(x)}{u(x) \ln(8)}$

STEP 9

Substitute $u(x)$ and $u'(x)$ into the formula for $y'$ :
$y' = \frac{-1 - 5x^4}{(9 - x - x^5) \ln(8)}$

STEP 10

Since the derivative should be expressed as a positive value, we can multiply the numerator and denominator by -1 to get:
$y' = \frac{1 + 5x^4}{(x^5 + x - 9) \ln(8)}$

STEP 11

Now we can match the expression for $y'$ with the given options. The correct derivative is:
$y' = \frac{5x^4 + 1}{(x^5 + x - 9) \ln(8)}$
This matches the first option given: $\frac{5 x^{4}+1}{\ln 8\left(x^{5}+x-9\right)}$

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Find the derivative of $y=\log_{8}(9-x-x^{5})$ .

STEP 1

Assumptions
1. We are given the function $y=\log _{8}\left(9-x-x^{5}\right)$ .
2. We need to find the derivative of $y$ with respect to $x$ , denoted as $y'$ .
3. We will use the chain rule and the logarithmic differentiation rule to find $y'$ .

STEP 2

The derivative of a logarithm with base $a$ is given by the formula:
$\frac{d}{dx} \log_a(u(x)) = \frac{u'(x)}{u(x) \ln(a)}$
where $u(x)$ is a differentiable function of $x$ .

STEP 3

Identify $u(x)$ in the given function:
$u(x) = 9 - x - x^5$

STEP 4

Find the derivative of $u(x)$ with respect to $x$ , denoted as $u'(x)$ :
$u'(x) = \frac{d}{dx}(9 - x - x^5)$

STEP 5

Differentiate each term of $u(x)$ separately:
$u'(x) = \frac{d}{dx}(9) - \frac{d}{dx}(x) - \frac{d}{dx}(x^5)$

STEP 6

Calculate the derivatives:
$u'(x) = 0 - 1 - 5x^4$

STEP 7

Simplify the expression for $u'(x)$ :
$u'(x) = -1 - 5x^4$

STEP 8

Now apply the logarithmic differentiation rule to find $y'$ :
$y' = \frac{u'(x)}{u(x) \ln(8)}$

STEP 9

Substitute $u(x)$ and $u'(x)$ into the formula for $y'$ :
$y' = \frac{-1 - 5x^4}{(9 - x - x^5) \ln(8)}$

STEP 10

Since the derivative should be expressed as a positive value, we can multiply the numerator and denominator by -1 to get:
$y' = \frac{1 + 5x^4}{(x^5 + x - 9) \ln(8)}$

STEP 11

Now we can match the expression for $y'$ with the given options. The correct derivative is:
$y' = \frac{5x^4 + 1}{(x^5 + x - 9) \ln(8)}$
This matches the first option given: $\frac{5 x^{4}+1}{\ln 8\left(x^{5}+x-9\right)}$

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Find the derivative of y=log⁡8(9−x−x5)y=\log_{8}(9-x-x^{5})y=log8​(9−x−x5).

STEP 1

Assumptions1. We are given the function y=log⁡8(9−x−x5)y=\log _{8}\left(9-x-x^{5}\right)y=log8​(9−x−x5).2. We need to find the derivative of yyy with respect to xxx, denoted as y′y'y′.3. We will use the chain rule and the logarithmic differentiation rule to find y′y'y′.

STEP 2

The derivative of a logarithm with base aaa is given by the formula:ddxlog⁡a(u(x))=u′(x)u(x)ln⁡(a)\frac{d}{dx} \log_a(u(x)) = \frac{u'(x)}{u(x) \ln(a)}dxd​loga​(u(x))=u(x)ln(a)u′(x)​where u(x)u(x)u(x) is a differentiable function of xxx.

STEP 3

Identify u(x)u(x)u(x) in the given function:u(x)=9−x−x5u(x) = 9 - x - x^5u(x)=9−x−x5

STEP 4

Find the derivative of u(x)u(x)u(x) with respect to xxx, denoted as u′(x)u'(x)u′(x):u′(x)=ddx(9−x−x5)u'(x) = \frac{d}{dx}(9 - x - x^5)u′(x)=dxd​(9−x−x5)

STEP 5

Differentiate each term of u(x)u(x)u(x) separately:u′(x)=ddx(9)−ddx(x)−ddx(x5)u'(x) = \frac{d}{dx}(9) - \frac{d}{dx}(x) - \frac{d}{dx}(x^5)u′(x)=dxd​(9)−dxd​(x)−dxd​(x5)

STEP 6

Calculate the derivatives:u′(x)=0−1−5x4u'(x) = 0 - 1 - 5x^4u′(x)=0−1−5x4

STEP 7

Simplify the expression for u′(x)u'(x)u′(x):u′(x)=−1−5x4u'(x) = -1 - 5x^4u′(x)=−1−5x4

STEP 8

Now apply the logarithmic differentiation rule to find y′y'y′:y′=u′(x)u(x)ln⁡(8)y' = \frac{u'(x)}{u(x) \ln(8)}y′=u(x)ln(8)u′(x)​

STEP 9

Substitute u(x)u(x)u(x) and u′(x)u'(x)u′(x) into the formula for y′y'y′:y′=−1−5x4(9−x−x5)ln⁡(8)y' = \frac{-1 - 5x^4}{(9 - x - x^5) \ln(8)}y′=(9−x−x5)ln(8)−1−5x4​

STEP 10

Since the derivative should be expressed as a positive value, we can multiply the numerator and denominator by -1 to get:y′=1+5x4(x5+x−9)ln⁡(8)y' = \frac{1 + 5x^4}{(x^5 + x - 9) \ln(8)}y′=(x5+x−9)ln(8)1+5x4​

STEP 11

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Find the derivative of $y=\log_{8}(9-x-x^{5})$ .

Assumptions
1. We are given the function $y=\log _{8}\left(9-x-x^{5}\right)$ .
2. We need to find the derivative of $y$ with respect to $x$ , denoted as $y'$ .
3. We will use the chain rule and the logarithmic differentiation rule to find $y'$ .

The derivative of a logarithm with base $a$ is given by the formula:
$\frac{d}{dx} \log_a(u(x)) = \frac{u'(x)}{u(x) \ln(a)}$
where $u(x)$ is a differentiable function of $x$ .

Identify $u(x)$ in the given function:
$u(x) = 9 - x - x^5$

Find the derivative of $u(x)$ with respect to $x$ , denoted as $u'(x)$ :
$u'(x) = \frac{d}{dx}(9 - x - x^5)$

Differentiate each term of $u(x)$ separately:
$u'(x) = \frac{d}{dx}(9) - \frac{d}{dx}(x) - \frac{d}{dx}(x^5)$

Calculate the derivatives:
$u'(x) = 0 - 1 - 5x^4$

Simplify the expression for $u'(x)$ :
$u'(x) = -1 - 5x^4$

Now apply the logarithmic differentiation rule to find $y'$ :
$y' = \frac{u'(x)}{u(x) \ln(8)}$

Substitute $u(x)$ and $u'(x)$ into the formula for $y'$ :
$y' = \frac{-1 - 5x^4}{(9 - x - x^5) \ln(8)}$

Since the derivative should be expressed as a positive value, we can multiply the numerator and denominator by -1 to get:
$y' = \frac{1 + 5x^4}{(x^5 + x - 9) \ln(8)}$