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Derivative of e^(sin(x)-2*x^3)
Identical expressions
e^(sin(x)- two *x^ three)
e to the power of ( sinus of (x) minus 2 multiply by x cubed )
e to the power of ( sinus of (x) minus two multiply by x to the power of three)
e(sin(x)-2*x3)
esinx-2*x3
e^(sin(x)-2*x³)
e to the power of (sin(x)-2*x to the power of 3)
e^(sin(x)-2x^3)
e(sin(x)-2x3)
esinx-2x3
e^sinx-2x^3
Similar expressions
e^(sin(x)+2*x^3)
e^(sinx-2*x^3)

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

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

The solution

You have entered [src]

             3
 sin(x) - 2*x 
E

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

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

Detail solution

Let $u = - 2 x^{3} + \sin{\left(x \right)}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- 2 x^{3} + \sin{\left(x \right)}\right)$ :
1. Differentiate $- 2 x^{3} + \sin{\left(x \right)}$ 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: $x^{3}$ goes to $3 x^{2}$
    So, the result is: $- 6 x^{2}$
  The result is: $- 6 x^{2} + \cos{\left(x \right)}$
The result of the chain rule is:

$\left(- 6 x^{2} + \cos{\left(x \right)}\right) e^{- 2 x^{3} + \sin{\left(x \right)}}$
Now simplify:

$\left(- 6 x^{2} + \cos{\left(x \right)}\right) e^{- 2 x^{3} + \sin{\left(x \right)}}$

The answer is:

$\left(- 6 x^{2} + \cos{\left(x \right)}\right) e^{- 2 x^{3} + \sin{\left(x \right)}}$

The first derivative [src]

                               3
/     2         \  sin(x) - 2*x 
\- 6*x  + cos(x)/*e

$$\left(- 6 x^{2} + \cos{\left(x \right)}\right) e^{- 2 x^{3} + \sin{\left(x \right)}}$$

Simplify

The second derivative [src]

/                2                \       3         
|/             2\                 |  - 2*x  + sin(x)
\\-cos(x) + 6*x /  - sin(x) - 12*x/*e

$$\left(- 12 x + \left(6 x^{2} - \cos{\left(x \right)}\right)^{2} - \sin{\left(x \right)}\right) e^{- 2 x^{3} + \sin{\left(x \right)}}$$

Simplify

The third derivative [src]

/                      3                                              \       3         
|      /             2\               /             2\                |  - 2*x  + sin(x)
\-12 - \-cos(x) + 6*x /  - cos(x) + 3*\-cos(x) + 6*x /*(12*x + sin(x))/*e

$$\left(3 \left(12 x + \sin{\left(x \right)}\right) \left(6 x^{2} - \cos{\left(x \right)}\right) - \left(6 x^{2} - \cos{\left(x \right)}\right)^{3} - \cos{\left(x \right)} - 12\right) e^{- 2 x^{3} + \sin{\left(x \right)}}$$

Simplify

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

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