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sin(e^x)
The derivative of the function/ sin(e^x)

Derivative of sin(e^x)

Function f() - derivative -N order at the point
v

The graph:

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The solution

You have entered [src] 
   / x\
sin\e /
sin⁡(ex)\sin{\left(e^{x} \right)} 
d /   / x\\
--\sin\e //
dx
ddxsin⁡(ex)\frac{d}{d x} \sin{\left(e^{x} \right)}
Transform
Detail solution

  1. Let u=exu = e^{x}.

  2. The derivative of sine is cosine:

    ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

  3. Then, apply the chain rule. Multiply by ddxex\frac{d}{d x} e^{x}:

    1. The derivative of exe^{x} is itself.

    The result of the chain rule is:

    excos⁡(ex)e^{x} \cos{\left(e^{x} \right)}

  4. Now simplify:

    excos⁡(ex)e^{x} \cos{\left(e^{x} \right)}

The answer is:

excos⁡(ex)e^{x} \cos{\left(e^{x} \right)}

The graph
02468-8-6-4-2-1010-2000020000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   / x\  x
cos\e /*e
excos⁡(ex)e^{x} \cos{\left(e^{x} \right)}
Simplify
The second derivative [src] 
/   x    / x\      / x\\  x
\- e *sin\e / + cos\e //*e
(−exsin⁡(ex)+cos⁡(ex))ex\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
(−e2xcos⁡(ex)−3exsin⁡(ex)+cos⁡(ex))ex\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
Derivative of sin(e^x)
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