Derivative of sin(e^x)

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   / x\
sin\e /

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

d /   / x\\
--\sin\e //
dx

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

Transform

Detail solution

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

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   / x\  x
cos\e /*e

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph