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e^(sin(x)- two *x^ two)
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Similar expressions
e^(sin(x)+2*x^2)
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The derivative of the function/ e^(sin(x)-2*x^2)

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

The solution

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             2
 sin(x) - 2*x 
E

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

E^(sin(x) - 2*x^2)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                             2
                 sin(x) - 2*x 
(-4*x + cos(x))*e

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                                     2         
/                 3                                          \  - 2*x  + sin(x)
\- (-cos(x) + 4*x)  - cos(x) + 3*(4 + sin(x))*(-cos(x) + 4*x)/*e

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

Simplify

The graph

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

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The solution