Derivative of cos(e^x)

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   / x\
cos\E /

$$\cos{\left(e^{x} \right)}$$

cos(E^x)

Detail solution

Let $u = e^{x}$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} e^{x}$ :
1. The derivative of $e^{x}$ is itself.
The result of the chain rule is:

$- e^{x} \sin{\left(e^{x} \right)}$
Now simplify:

$- e^{x} \sin{\left(e^{x} \right)}$

The answer is:

$- e^{x} \sin{\left(e^{x} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  x    / x\
-e *sin\E /

$$- e^{x} \sin{\left(e^{x} \right)}$$

Simplify

The second derivative [src]

 /   / x\  x      / x\\  x
-\cos\E /*e  + sin\E //*e

$$- \left(e^{x} \cos{\left(e^{x} \right)} + \sin{\left(e^{x} \right)}\right) e^{x}$$

Simplify

The third derivative [src]

/     / x\    2*x    / x\        / x\  x\  x
\- sin\E / + e   *sin\E / - 3*cos\E /*e /*e

$$\left(e^{2 x} \sin{\left(e^{x} \right)} - 3 e^{x} \cos{\left(e^{x} \right)} - \sin{\left(e^{x} \right)}\right) e^{x}$$

Simplify

The graph