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Derivative of y=e^(cosx^2)
Identical expressions
y=e^(cosx^ two)
y equally e to the power of ( co sinus of e of x squared )
y equally e to the power of ( co sinus of e of x to the power of two)
y=e(cosx2)
y=ecosx2
y=e^(cosx²)
y=e to the power of (cosx to the power of 2)
y=e^cosx^2
Similar expressions
e^cosx^2

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

Derivative of y=e^(cosx^2)

The solution

You have entered [src]

    2   
 cos (x)
e

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

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

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

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)

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

Simplify

The graph

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Derivative of y=e^(cosx^2)

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Piecewise:

The solution