Derivative of sinxe^cosx

The solution

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        cos(x)
sin(x)*E

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

sin(x)*E^cos(x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
$g{\left(x \right)} = e^{\cos{\left(x \right)}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
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:
  
  $- e^{\cos{\left(x \right)}} \sin{\left(x \right)}$
The result is: $- e^{\cos{\left(x \right)}} \sin^{2}{\left(x \right)} + e^{\cos{\left(x \right)}} \cos{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        cos(x)      2     cos(x)
cos(x)*e       - sin (x)*e

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

Simplify

The second derivative [src]

/        2              \  cos(x)       
\-1 + sin (x) - 3*cos(x)/*e      *sin(x)

$$\left(\sin^{2}{\left(x \right)} - 3 \cos{\left(x \right)} - 1\right) e^{\cos{\left(x \right)}} \sin{\left(x \right)}$$

Simplify

The third derivative [src]

/               2         2    /       2              \     /   2            \       \  cos(x)
\-cos(x) + 3*sin (x) + sin (x)*\1 - sin (x) + 3*cos(x)/ + 3*\sin (x) - cos(x)/*cos(x)/*e

$$\left(3 \left(\sin^{2}{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(x \right)} + \left(- \sin^{2}{\left(x \right)} + 3 \cos{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} - \cos{\left(x \right)}\right) e^{\cos{\left(x \right)}}$$

Simplify

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Identical expressions

Derivative of sinxe^cosx

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The solution