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e^(three *sin(x)- two x^2)
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Similar expressions
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The derivative of the function/ e^(3*sin(x)-2x^2)

Derivative of e^(3*sin(x)-2x^2)

The solution

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

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

E^(3*sin(x) - 2*x^2)

Detail solution

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

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of e^(3*sin(x)-2x^2)

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The solution