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

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

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
    2   
 sin (x)
e
esin⁡2(x)e^{\sin^{2}{\left(x \right)}} 
  /    2   \
d | sin (x)|
--\e       /
dx
ddxesin⁡2(x)\frac{d}{d x} e^{\sin^{2}{\left(x \right)}}
Transform
Detail solution

  1. Let u=sin⁡2(x)u = \sin^{2}{\left(x \right)}.

  2. The derivative of eue^{u} is itself.

  3. Then, apply the chain rule. Multiply by ddxsin⁡2(x)\frac{d}{d x} \sin^{2}{\left(x \right)}:

    1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

    2. Apply the power rule: u2u^{2} goes to 2u2 u

    3. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\left(x \right)}:

      1. The derivative of sine is cosine:

        ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

      The result of the chain rule is:

      2sin⁡(x)cos⁡(x)2 \sin{\left(x \right)} \cos{\left(x \right)}

    The result of the chain rule is:

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

  4. Now simplify:

    e12−cos⁡(2x)2sin⁡(2x)e^{\frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}} \sin{\left(2 x \right)}

The answer is:

e12−cos⁡(2x)2sin⁡(2x)e^{\frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}} \sin{\left(2 x \right)}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
             2          
          sin (x)       
2*cos(x)*e       *sin(x)
2esin⁡2(x)sin⁡(x)cos⁡(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⋅(2sin⁡2(x)cos⁡2(x)−sin⁡2(x)+cos⁡2(x))esin⁡2(x)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⋅(2sin⁡2(x)cos⁡2(x)−3sin⁡2(x)+3cos⁡2(x)−2)esin⁡2(x)sin⁡(x)cos⁡(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
Derivative of e^sin(x)^(2)
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