Derivative of exp^(xcosx)

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 x*cos(x)
e

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

d / x*cos(x)\
--\e        /
dx

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

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Detail solution

Let $u = x \cos{\left(x \right)}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} x \cos{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result is: $- x \sin{\left(x \right)} + \cos{\left(x \right)}$
The result of the chain rule is:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

                      x*cos(x)
(-x*sin(x) + cos(x))*e

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of exp^(xcosx)

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The solution