Derivative of е^sinx*sinx

The solution

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 sin(x)       
E      *sin(x)

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

E^sin(x)*sin(x)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

 /         2                 /       2              \       \         sin(x)
-\1 - 3*cos (x) + 6*sin(x) + \1 - cos (x) + 3*sin(x)/*sin(x)/*cos(x)*e

$$- \left(\left(3 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + 6 \sin{\left(x \right)} - 3 \cos^{2}{\left(x \right)} + 1\right) e^{\sin{\left(x \right)}} \cos{\left(x \right)}$$

Simplify

The graph

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Identical expressions

Derivative of е^sinx*sinx

The graph:

Piecewise:

The solution