Solved on Feb 24, 2024

Find the derivative of $\sin(g(x))$ evaluated at $x=8$ , given that $g(8)=3$ and $g'(8)=9$ . Round the answer to three decimal places.

STEP 1

Assumptions
1. The function $g$ is differentiable.
2. We are given $g(8) = 3$ and $g'(8) = 9$ .
3. We need to compute the derivative of $\sin(g(x))$ with respect to $x$ .
4. The evaluation of the derivative will be at $x = 8$ .
5. The answer should be in decimal form, rounded to three decimal places.
6. Calculations are to be done assuming the calculator is in radian mode.

STEP 2

To find the derivative of $\sin(g(x))$ with respect to $x$ , we will use the chain rule. The chain rule states that if we have a composite function $f(g(x))$ , then the derivative $f'(g(x))$ is $f'(g(x)) \cdot g'(x)$ .
$\frac{d}{dx}[\sin(g(x))] = \cos(g(x)) \cdot g'(x)$

STEP 3

Now we need to evaluate the derivative at $x = 8$ .
$\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} = \cos(g(8)) \cdot g'(8)$

STEP 4

Substitute the given values $g(8) = 3$ and $g'(8) = 9$ into the derivative.
$\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} = \cos(3) \cdot 9$

STEP 5

Calculate the value of $\cos(3)$ using a calculator set to radian mode. Round the result to at least three decimal places to ensure accuracy in the final answer.
$\cos(3) \approx 0.990$

STEP 6

Now, multiply the value of $\cos(3)$ by $g'(8)$ to find the derivative evaluated at $x = 8$ .
$\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} \approx 0.990 \cdot 9$

STEP 7

Calculate the final result and round it to three decimal places.
$\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} \approx 8.910$
The derivative of $\sin(g(x))$ evaluated at $x = 8$ is approximately $8.910$ .

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Find the derivative of $\sin(g(x))$ evaluated at $x=8$ , given that $g(8)=3$ and $g'(8)=9$ . Round the answer to three decimal places.

STEP 1

STEP 2

To find the derivative of $\sin(g(x))$ with respect to $x$ , we will use the chain rule. The chain rule states that if we have a composite function $f(g(x))$ , then the derivative $f'(g(x))$ is $f'(g(x)) \cdot g'(x)$ .
$\frac{d}{dx}[\sin(g(x))] = \cos(g(x)) \cdot g'(x)$

STEP 3

Now we need to evaluate the derivative at $x = 8$ .
$\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} = \cos(g(8)) \cdot g'(8)$

STEP 4

Substitute the given values $g(8) = 3$ and $g'(8) = 9$ into the derivative.
$\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} = \cos(3) \cdot 9$

STEP 5

Calculate the value of $\cos(3)$ using a calculator set to radian mode. Round the result to at least three decimal places to ensure accuracy in the final answer.
$\cos(3) \approx 0.990$

STEP 6

Now, multiply the value of $\cos(3)$ by $g'(8)$ to find the derivative evaluated at $x = 8$ .
$\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} \approx 0.990 \cdot 9$

STEP 7

Calculate the final result and round it to three decimal places.
$\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} \approx 8.910$
The derivative of $\sin(g(x))$ evaluated at $x = 8$ is approximately $8.910$ .

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Find the derivative of sin⁡(g(x))\sin(g(x))sin(g(x)) evaluated at x=8x=8x=8, given that g(8)=3g(8)=3g(8)=3 and g′(8)=9g'(8)=9g′(8)=9. Round the answer to three decimal places.

STEP 1

STEP 2

STEP 3

Now we need to evaluate the derivative at x=8x = 8x=8.ddx[sin⁡(g(x))]∣x=8=cos⁡(g(8))⋅g′(8)\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} = \cos(g(8)) \cdot g'(8)dxd​[sin(g(x))]∣∣​x=8​=cos(g(8))⋅g′(8)

STEP 4

Substitute the given values g(8)=3g(8) = 3g(8)=3 and g′(8)=9g'(8) = 9g′(8)=9 into the derivative.ddx[sin⁡(g(x))]∣x=8=cos⁡(3)⋅9\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} = \cos(3) \cdot 9dxd​[sin(g(x))]∣∣​x=8​=cos(3)⋅9

STEP 5

Calculate the value of cos⁡(3)\cos(3)cos(3) using a calculator set to radian mode. Round the result to at least three decimal places to ensure accuracy in the final answer.cos⁡(3)≈0.990\cos(3) \approx 0.990cos(3)≈0.990

STEP 6

Now, multiply the value of cos⁡(3)\cos(3)cos(3) by g′(8)g'(8)g′(8) to find the derivative evaluated at x=8x = 8x=8.ddx[sin⁡(g(x))]∣x=8≈0.990⋅9\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} \approx 0.990 \cdot 9dxd​[sin(g(x))]∣∣​x=8​≈0.990⋅9

STEP 7

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Find the derivative of $\sin(g(x))$ evaluated at $x=8$ , given that $g(8)=3$ and $g'(8)=9$ . Round the answer to three decimal places.

Now we need to evaluate the derivative at $x = 8$ .
$\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} = \cos(g(8)) \cdot g'(8)$

Substitute the given values $g(8) = 3$ and $g'(8) = 9$ into the derivative.
$\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} = \cos(3) \cdot 9$

Calculate the value of $\cos(3)$ using a calculator set to radian mode. Round the result to at least three decimal places to ensure accuracy in the final answer.
$\cos(3) \approx 0.990$

Now, multiply the value of $\cos(3)$ by $g'(8)$ to find the derivative evaluated at $x = 8$ .
$\left.\frac{d}{dx}[\sin(g(x))]\right|_{x=8} \approx 0.990 \cdot 9$