Solved on Mar 17, 2024

Compose and differentiate $y = e^{5\sqrt{x}}$ . Identify the inner function $u=g(x)$ and the outer function $y=f(u)$ .
$(f(u), g(x)) = (e^u, \sqrt{x})$
$\frac{dy}{dx} = 25e^{5\sqrt{x}}\frac{1}{2\sqrt{x}}$

STEP 1

Assumptions
1. We are given a composite function $y = e^{5 \sqrt{x}}$ .
2. We need to identify the inner function $u = g(x)$ and the outer function $y = f(u)$ .
3. We need to find the derivative $\frac{dy}{dx}$ of the composite function.

STEP 2

Identify the inner function $g(x)$ , which is the part inside the composition that is applied first when evaluating the function. In this case, the inner function is the square root function applied to $x$ , multiplied by 5.
$u = g(x) = 5 \sqrt{x}$

STEP 3

Identify the outer function $f(u)$ , which is the function that takes the result of the inner function as its input. In this case, the outer function is the exponential function.
$y = f(u) = e^u$

STEP 4

Write the composite function in the form $f(g(x))$ using the identified inner and outer functions.
$y = f(g(x)) = e^{5 \sqrt{x}}$

STEP 5

Now we will find the derivative of the outer function $f(u)$ with respect to $u$ .
$\frac{df}{du} = \frac{d}{du}(e^u) = e^u$

STEP 6

Next, we find the derivative of the inner function $g(x)$ with respect to $x$ .
$\frac{dg}{dx} = \frac{d}{dx}(5 \sqrt{x})$

STEP 7

To differentiate $5 \sqrt{x}$ , we rewrite it as $5x^{1/2}$ .
$\frac{dg}{dx} = \frac{d}{dx}(5x^{1/2})$

STEP 8

Apply the power rule for differentiation to find $\frac{dg}{dx}$ .
$\frac{dg}{dx} = 5 \cdot \frac{1}{2} x^{-1/2}$

STEP 9

Simplify the expression for $\frac{dg}{dx}$ .
$\frac{dg}{dx} = \frac{5}{2} x^{-1/2}$

STEP 10

Now, apply the chain rule for differentiation, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
$\frac{dy}{dx} = \frac{df}{du} \bigg|_{u=g(x)} \cdot \frac{dg}{dx}$

STEP 11

Substitute the expressions for $\frac{df}{du}$ and $\frac{dg}{dx}$ into the chain rule formula.
$\frac{dy}{dx} = e^{5 \sqrt{x}} \cdot \frac{5}{2} x^{-1/2}$

STEP 12

Simplify the expression for $\frac{dy}{dx}$ .
$\frac{dy}{dx} = \frac{5}{2} e^{5 \sqrt{x}} x^{-1/2}$
The derivative $\frac{dy}{dx}$ of the composite function $y = e^{5 \sqrt{x}}$ is $\frac{5}{2} e^{5 \sqrt{x}} x^{-1/2}$ .

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Compose and differentiate $y = e^{5\sqrt{x}}$ . Identify the inner function $u=g(x)$ and the outer function $y=f(u)$ .
$(f(u), g(x)) = (e^u, \sqrt{x})$
$\frac{dy}{dx} = 25e^{5\sqrt{x}}\frac{1}{2\sqrt{x}}$

STEP 1

Assumptions
1. We are given a composite function $y = e^{5 \sqrt{x}}$ .
2. We need to identify the inner function $u = g(x)$ and the outer function $y = f(u)$ .
3. We need to find the derivative $\frac{dy}{dx}$ of the composite function.

STEP 2

Identify the inner function $g(x)$ , which is the part inside the composition that is applied first when evaluating the function. In this case, the inner function is the square root function applied to $x$ , multiplied by 5.
$u = g(x) = 5 \sqrt{x}$

STEP 3

Identify the outer function $f(u)$ , which is the function that takes the result of the inner function as its input. In this case, the outer function is the exponential function.
$y = f(u) = e^u$

STEP 4

Write the composite function in the form $f(g(x))$ using the identified inner and outer functions.
$y = f(g(x)) = e^{5 \sqrt{x}}$

STEP 5

Now we will find the derivative of the outer function $f(u)$ with respect to $u$ .
$\frac{df}{du} = \frac{d}{du}(e^u) = e^u$

STEP 6

Next, we find the derivative of the inner function $g(x)$ with respect to $x$ .
$\frac{dg}{dx} = \frac{d}{dx}(5 \sqrt{x})$

STEP 7

To differentiate $5 \sqrt{x}$ , we rewrite it as $5x^{1/2}$ .
$\frac{dg}{dx} = \frac{d}{dx}(5x^{1/2})$

STEP 8

Apply the power rule for differentiation to find $\frac{dg}{dx}$ .
$\frac{dg}{dx} = 5 \cdot \frac{1}{2} x^{-1/2}$

STEP 9

Simplify the expression for $\frac{dg}{dx}$ .
$\frac{dg}{dx} = \frac{5}{2} x^{-1/2}$

STEP 10

Now, apply the chain rule for differentiation, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
$\frac{dy}{dx} = \frac{df}{du} \bigg|_{u=g(x)} \cdot \frac{dg}{dx}$

STEP 11

Substitute the expressions for $\frac{df}{du}$ and $\frac{dg}{dx}$ into the chain rule formula.
$\frac{dy}{dx} = e^{5 \sqrt{x}} \cdot \frac{5}{2} x^{-1/2}$

STEP 12

Simplify the expression for $\frac{dy}{dx}$ .
$\frac{dy}{dx} = \frac{5}{2} e^{5 \sqrt{x}} x^{-1/2}$
The derivative $\frac{dy}{dx}$ of the composite function $y = e^{5 \sqrt{x}}$ is $\frac{5}{2} e^{5 \sqrt{x}} x^{-1/2}$ .

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STEP 1

Assumptions1. We are given a composite function y=e5xy = e^{5 \sqrt{x}}y=e5x​.2. We need to identify the inner function u=g(x)u = g(x)u=g(x) and the outer function y=f(u)y = f(u)y=f(u).3. We need to find the derivative dydx\frac{dy}{dx}dxdy​ of the composite function.

STEP 2

Identify the inner function g(x)g(x)g(x), which is the part inside the composition that is applied first when evaluating the function. In this case, the inner function is the square root function applied to xxx, multiplied by 5.u=g(x)=5xu = g(x) = 5 \sqrt{x}u=g(x)=5x​

STEP 3

Identify the outer function f(u)f(u)f(u), which is the function that takes the result of the inner function as its input. In this case, the outer function is the exponential function.y=f(u)=euy = f(u) = e^uy=f(u)=eu

STEP 4

Write the composite function in the form f(g(x))f(g(x))f(g(x)) using the identified inner and outer functions.y=f(g(x))=e5xy = f(g(x)) = e^{5 \sqrt{x}}y=f(g(x))=e5x​

STEP 5

Now we will find the derivative of the outer function f(u)f(u)f(u) with respect to uuu.dfdu=ddu(eu)=eu\frac{df}{du} = \frac{d}{du}(e^u) = e^ududf​=dud​(eu)=eu

STEP 6

Next, we find the derivative of the inner function g(x)g(x)g(x) with respect to xxx.dgdx=ddx(5x)\frac{dg}{dx} = \frac{d}{dx}(5 \sqrt{x})dxdg​=dxd​(5x​)

STEP 7

To differentiate 5x5 \sqrt{x}5x​, we rewrite it as 5x1/25x^{1/2}5x1/2.dgdx=ddx(5x1/2)\frac{dg}{dx} = \frac{d}{dx}(5x^{1/2})dxdg​=dxd​(5x1/2)

STEP 8

Apply the power rule for differentiation to find dgdx\frac{dg}{dx}dxdg​.dgdx=5⋅12x−1/2\frac{dg}{dx} = 5 \cdot \frac{1}{2} x^{-1/2}dxdg​=5⋅21​x−1/2

STEP 9

Simplify the expression for dgdx\frac{dg}{dx}dxdg​.dgdx=52x−1/2\frac{dg}{dx} = \frac{5}{2} x^{-1/2}dxdg​=25​x−1/2

STEP 10

STEP 11

Substitute the expressions for dfdu\frac{df}{du}dudf​ and dgdx\frac{dg}{dx}dxdg​ into the chain rule formula.dydx=e5x⋅52x−1/2\frac{dy}{dx} = e^{5 \sqrt{x}} \cdot \frac{5}{2} x^{-1/2}dxdy​=e5x​⋅25​x−1/2

STEP 12

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Assumptions
1. We are given a composite function $y = e^{5 \sqrt{x}}$ .
2. We need to identify the inner function $u = g(x)$ and the outer function $y = f(u)$ .
3. We need to find the derivative $\frac{dy}{dx}$ of the composite function.

Identify the inner function $g(x)$ , which is the part inside the composition that is applied first when evaluating the function. In this case, the inner function is the square root function applied to $x$ , multiplied by 5.
$u = g(x) = 5 \sqrt{x}$

Identify the outer function $f(u)$ , which is the function that takes the result of the inner function as its input. In this case, the outer function is the exponential function.
$y = f(u) = e^u$

Write the composite function in the form $f(g(x))$ using the identified inner and outer functions.
$y = f(g(x)) = e^{5 \sqrt{x}}$

Now we will find the derivative of the outer function $f(u)$ with respect to $u$ .
$\frac{df}{du} = \frac{d}{du}(e^u) = e^u$

Next, we find the derivative of the inner function $g(x)$ with respect to $x$ .
$\frac{dg}{dx} = \frac{d}{dx}(5 \sqrt{x})$

To differentiate $5 \sqrt{x}$ , we rewrite it as $5x^{1/2}$ .
$\frac{dg}{dx} = \frac{d}{dx}(5x^{1/2})$

Apply the power rule for differentiation to find $\frac{dg}{dx}$ .
$\frac{dg}{dx} = 5 \cdot \frac{1}{2} x^{-1/2}$

Simplify the expression for $\frac{dg}{dx}$ .
$\frac{dg}{dx} = \frac{5}{2} x^{-1/2}$

Substitute the expressions for $\frac{df}{du}$ and $\frac{dg}{dx}$ into the chain rule formula.
$\frac{dy}{dx} = e^{5 \sqrt{x}} \cdot \frac{5}{2} x^{-1/2}$