Mister Exam

Derivative of cos(e^x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   / x\
cos\E /
cos(ex)\cos{\left(e^{x} \right)}
cos(E^x)
Detail solution
  1. Let u=exu = e^{x}.

  2. The derivative of cosine is negative sine:

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

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

  4. Now simplify:

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


The answer is:

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

The graph
02468-8-6-4-2-1010-5000050000
The first derivative [src]
  x    / x\
-e *sin\E /
exsin(ex)- e^{x} \sin{\left(e^{x} \right)}
The second derivative [src]
 /   / x\  x      / x\\  x
-\cos\E /*e  + sin\E //*e 
(excos(ex)+sin(ex))ex- \left(e^{x} \cos{\left(e^{x} \right)} + \sin{\left(e^{x} \right)}\right) e^{x}
The third derivative [src]
/     / x\    2*x    / x\        / x\  x\  x
\- sin\E / + e   *sin\E / - 3*cos\E /*e /*e 
(e2xsin(ex)3excos(ex)sin(ex))ex\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}
The graph
Derivative of cos(e^x)