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(e^(cosx)+2)^2

Derivative of (e^(cosx)+2)^2

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
             2
/ cos(x)    \ 
\e       + 2/ 
$$\left(e^{\cos{\left(x \right)}} + 2\right)^{2}$$
  /             2\
d |/ cos(x)    \ |
--\\e       + 2/ /
dx                
$$\frac{d}{d x} \left(e^{\cos{\left(x \right)}} + 2\right)^{2}$$
Detail solution
  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Differentiate term by term:

      1. Let .

      2. The derivative of is itself.

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of cosine is negative sine:

        The result of the chain rule is:

      4. The derivative of the constant is zero.

      The result is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
   / cos(x)    \  cos(x)       
-2*\e       + 2/*e      *sin(x)
$$- 2 \left(e^{\cos{\left(x \right)}} + 2\right) e^{\cos{\left(x \right)}} \sin{\left(x \right)}$$
The second derivative [src]
  /   2    /     cos(x)\      2     cos(x)   /     cos(x)\       \  cos(x)
2*\sin (x)*\2 + e      / + sin (x)*e       - \2 + e      /*cos(x)/*e      
$$2 \left(\left(e^{\cos{\left(x \right)}} + 2\right) \sin^{2}{\left(x \right)} + e^{\cos{\left(x \right)}} \sin^{2}{\left(x \right)} - \left(e^{\cos{\left(x \right)}} + 2\right) \cos{\left(x \right)}\right) e^{\cos{\left(x \right)}}$$
The third derivative [src]
  /       2    /     cos(x)\        2     cos(x)     /     cos(x)\                    cos(x)    cos(x)\  cos(x)       
2*\2 - sin (x)*\2 + e      / - 3*sin (x)*e       + 3*\2 + e      /*cos(x) + 3*cos(x)*e       + e      /*e      *sin(x)
$$2 \left(- \left(e^{\cos{\left(x \right)}} + 2\right) \sin^{2}{\left(x \right)} - 3 e^{\cos{\left(x \right)}} \sin^{2}{\left(x \right)} + 3 \left(e^{\cos{\left(x \right)}} + 2\right) \cos{\left(x \right)} + 3 e^{\cos{\left(x \right)}} \cos{\left(x \right)} + e^{\cos{\left(x \right)}} + 2\right) e^{\cos{\left(x \right)}} \sin{\left(x \right)}$$
The graph
Derivative of (e^(cosx)+2)^2