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Derivative of e^(2*cos(t)^(2)-2*sin(t)^(2))
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Similar expressions
e^(2*cos(t)^(2)+2*sin(t)^(2))

The derivative of the function/ e^(2*cos(t)^(2)-2*sin(t)^(2))

Derivative of e^(2cos(t)^(2)-2sin(t)^(2))

The solution

You have entered [src]

      2           2   
 2*cos (t) - 2*sin (t)
E

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

E^(2*cos(t)^2 - 2*sin(t)^2)

Detail solution

Let $u = - 2 \sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d t} \left(- 2 \sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}\right)$ :
1. Differentiate $- 2 \sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \cos{\left(t \right)}$ .
    2. Apply the power rule: $u^{2}$ goes to $2 u$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d t} \cos{\left(t \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
      The result of the chain rule is:
      
      $- 2 \sin{\left(t \right)} \cos{\left(t \right)}$
    So, the result is: $- 4 \sin{\left(t \right)} \cos{\left(t \right)}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \sin{\left(t \right)}$ .
    2. Apply the power rule: $u^{2}$ goes to $2 u$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d t} \sin{\left(t \right)}$ :
      1. The derivative of sine is cosine:
        
        $\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$
      The result of the chain rule is:
      
      $2 \sin{\left(t \right)} \cos{\left(t \right)}$
    So, the result is: $- 4 \sin{\left(t \right)} \cos{\left(t \right)}$
  The result is: $- 8 \sin{\left(t \right)} \cos{\left(t \right)}$
The result of the chain rule is:

$- 8 e^{- 2 \sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}} \sin{\left(t \right)} \cos{\left(t \right)}$
Now simplify:

$- 4 e^{2 \cos{\left(2 t \right)}} \sin{\left(2 t \right)}$

The answer is:

$- 4 e^{2 \cos{\left(2 t \right)}} \sin{\left(2 t \right)}$

The graph

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The first derivative [src]

                2           2          
           2*cos (t) - 2*sin (t)       
-8*cos(t)*e                     *sin(t)

$$- 8 e^{- 2 \sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}} \sin{\left(t \right)} \cos{\left(t \right)}$$

Simplify

The second derivative [src]

                                                  2           2   
  /   2         2           2       2   \  - 2*sin (t) + 2*cos (t)
8*\sin (t) - cos (t) + 8*cos (t)*sin (t)/*e

$$8 \left(8 \sin^{2}{\left(t \right)} \cos^{2}{\left(t \right)} + \sin^{2}{\left(t \right)} - \cos^{2}{\left(t \right)}\right) e^{- 2 \sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}}$$

Simplify

The third derivative [src]

                                                                   2           2          
   /         2           2            2       2   \         - 2*sin (t) + 2*cos (t)       
32*\1 - 6*sin (t) + 6*cos (t) - 16*cos (t)*sin (t)/*cos(t)*e                       *sin(t)

$$32 \left(- 16 \sin^{2}{\left(t \right)} \cos^{2}{\left(t \right)} - 6 \sin^{2}{\left(t \right)} + 6 \cos^{2}{\left(t \right)} + 1\right) e^{- 2 \sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}} \sin{\left(t \right)} \cos{\left(t \right)}$$

Simplify

The graph

Derivative of e^(2*cos(t)^(2)-2*sin(t)^(2))

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Derivative of e^(2cos(t)^(2)-2sin(t)^(2))

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Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of e^(2*cos(t)^(2)-2*sin(t)^(2))

The graph:

Piecewise:

The solution

Derivative of e^(2cos(t)^(2)-2sin(t)^(2))