Derivative of cos(e^x) by Chain Rule

The derivative of cos(e^x) is equal to -e^x sin(e^x). In this post, we will learn how to find the derivative of cos(e^x) by the chain rule of derivatives.

Derivative of cos(e^x)

Question: Find the derivative of cos(e^x).

Answer:

The derivative of cos(e^x) is equal to -e^xsin(e^x).

Explanation:

Note that f(x)=cos(e^x) is a composite function. The following steps are needed to find the derivative of cos(e^x) using the chain rule of derivatives.

Step 1: We assume that z=e^x

Step 2: We have that  $\dfrac{dz}{dx}=e^x$.

Step 3: Now, by the chain rule, the derivative of cos(e^x) is equal to

$\dfrac{d}{dx}(\cos e^x)=\dfrac{d}{dz}(\cos z) \cdot \dfrac{dz}{dx}$

= -sin z ⋅ e^x as we know that the derivative of sinx is cosx.

= -sin(e^x) ⋅ e^x, putting the value of z=e^x.

Conclusion: Hence, the derivative of cos(e^x) by the chain rule is -e^x sin(e^x).

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Question: What is the derivative of cos(e^x) at x=0?

Answer:

From the above, we have obtained that the derivative of cos(e^x) is -e^x sin(e^x). Thus, at the point x=0 we have that

$\dfrac{d}{dx}[\cos(e^x)]$

= [-e^x sin(e^x)]_{x=0}

= -e⁰ sin(e⁰)

= – 1 ⋅ sin 1

= -sin 1

Thus, the derivative of cos(e^x) at the point x=0 is -sin 1.

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