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e^ two ^x*sinx^ two
e squared to the power of x multiply by sinus of x squared
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e^2^xsinx^2
e2xsinx2

The derivative of the function/ e^2^x*sinx^2

Derivative of e^2^x*sinx^2

The solution

You have entered [src]

 / x\        
 \2 /    2   
e    *sin (x)

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

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

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

Transform

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                                                                                     / x\
/       2           2       x    2       2    /     x\      x                     \  \2 /
\- 2*sin (x) + 2*cos (x) + 2 *log (2)*sin (x)*\1 + 2 / + 4*2 *cos(x)*log(2)*sin(x)/*e

$$\left(2^{x} \left(2^{x} + 1\right) \log{\left(2 \right)}^{2} \sin^{2}{\left(x \right)} + 4 \cdot 2^{x} \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} - 2 \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right) e^{2^{x}}$$

Simplify

The third derivative [src]

                                                                                                                                   / x\
/                      x /   2         2   \           x    3       2    /     2*x      x\      x    2    /     x\              \  \2 /
\-8*cos(x)*sin(x) - 6*2 *\sin (x) - cos (x)/*log(2) + 2 *log (2)*sin (x)*\1 + 2    + 3*2 / + 6*2 *log (2)*\1 + 2 /*cos(x)*sin(x)/*e

$$\left(6 \cdot 2^{x} \left(2^{x} + 1\right) \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} - 6 \cdot 2^{x} \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} + 2^{x} \left(2^{2 x} + 3 \cdot 2^{x} + 1\right) \log{\left(2 \right)}^{3} \sin^{2}{\left(x \right)} - 8 \sin{\left(x \right)} \cos{\left(x \right)}\right) e^{2^{x}}$$

Simplify

The graph

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Identical expressions

Derivative of e^2^x*sinx^2

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The solution