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(e^(cosx)+ two)^ two
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Similar expressions
(e^(cosx)-2)^2

The derivative of the function/ (e^(cosx)+2)^2

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

The solution

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             2
/ cos(x)    \ 
\e       + 2/

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

  /             2\
d |/ cos(x)    \ |
--\\e       + 2/ /
dx

$$\frac{d}{d x} \left(e^{\cos{\left(x \right)}} + 2\right)^{2}$$

Transform

Detail solution

Let $u = e^{\cos{\left(x \right)}} + 2$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(e^{\cos{\left(x \right)}} + 2\right)$ :
1. Differentiate $e^{\cos{\left(x \right)}} + 2$ term by term:
  1. Let $u = \cos{\left(x \right)}$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result of the chain rule is:
    
    $- e^{\cos{\left(x \right)}} \sin{\left(x \right)}$
  4. The derivative of the constant $2$ is zero.
  The result is: $- e^{\cos{\left(x \right)}} \sin{\left(x \right)}$
The result of the chain rule is:

$- \left(2 e^{\cos{\left(x \right)}} + 4\right) e^{\cos{\left(x \right)}} \sin{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

   / cos(x)    \  cos(x)       
-2*\e       + 2/*e      *sin(x)

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

Simplify

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

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

Simplify

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)

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

Simplify

The graph

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

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The solution