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e^sinx^(2)

The derivative of the function/ e^sin(x)^(2)

Derivative of e^sin(x)^(2)

The solution

You have entered [src]

    2   
 sin (x)
e

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

  /    2   \
d | sin (x)|
--\e       /
dx

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

Transform

Detail solution

Let $u = \sin^{2}{\left(x \right)}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin^{2}{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
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:
  
  $2 \sin{\left(x \right)} \cos{\left(x \right)}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                              2          
  /          2           2           2       2   \         sin (x)       
4*\-2 - 3*sin (x) + 3*cos (x) + 2*cos (x)*sin (x)/*cos(x)*e       *sin(x)

$$4 \cdot \left(2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - 3 \sin^{2}{\left(x \right)} + 3 \cos^{2}{\left(x \right)} - 2\right) e^{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The graph

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Derivative of e^sin(x)^(2)

The graph:

Piecewise:

The solution