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y=e^(cosx^2)
The derivative of the function/ y=e^(cosx^2)

Derivative of y=e^(cosx^2)

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

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

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

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

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

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

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

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

      1. The derivative of cosine is negative sine:

        ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\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:

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

  4. Now simplify:

    −ecos⁡(2x)2+12sin⁡(2x)- e^{\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}} \sin{\left(2 x \right)}

The answer is:

−ecos⁡(2x)2+12sin⁡(2x)- e^{\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{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          
           cos (x)       
-2*cos(x)*e       *sin(x)
−2ecos⁡2(x)sin⁡(x)cos⁡(x)- 2 e^{\cos^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}
Simplify
The second derivative [src] 
                                              2   
  /   2         2           2       2   \  cos (x)
2*\sin (x) - cos (x) + 2*cos (x)*sin (x)/*e
2⋅(2sin⁡2(x)cos⁡2(x)+sin⁡2(x)−cos⁡2(x))ecos⁡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^{\cos^{2}{\left(x \right)}}
Simplify
The third derivative [src] 
                                                             2          
  /         2           2           2       2   \         cos (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)ecos⁡2(x)sin⁡(x)cos⁡(x)4 \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^{\cos^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}
Simplify
The graph
Derivative of y=e^(cosx^2)
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