Derivative of y=(sin(5x²-6x+e^x))

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   /   2          x\
sin\5*x  - 6*x + E /

$$\sin{\left(e^{x} + \left(5 x^{2} - 6 x\right) \right)}$$

sin(5*x^2 - 6*x + E^x)

Detail solution

Let $u = e^{x} + \left(5 x^{2} - 6 x\right)$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(e^{x} + \left(5 x^{2} - 6 x\right)\right)$ :
1. Differentiate $e^{x} + \left(5 x^{2} - 6 x\right)$ term by term:
  1. Differentiate $5 x^{2} - 6 x$ term by term:
    1. 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: $10 x$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $-6$
    The result is: $10 x - 6$
  2. The derivative of $e^{x}$ is itself.
  The result is: $10 x + e^{x} - 6$
The result of the chain rule is:

$\left(10 x + e^{x} - 6\right) \cos{\left(e^{x} + \left(5 x^{2} - 6 x\right) \right)}$
Now simplify:

$\left(10 x + e^{x} - 6\right) \cos{\left(5 x^{2} - 6 x + e^{x} \right)}$

The answer is:

$\left(10 x + e^{x} - 6\right) \cos{\left(5 x^{2} - 6 x + e^{x} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/      x       \    /   2          x\
\-6 + E  + 10*x/*cos\5*x  - 6*x + E /

$$\left(e^{x} + 10 x - 6\right) \cos{\left(e^{x} + \left(5 x^{2} - 6 x\right) \right)}$$

Simplify

The second derivative [src]

                                                  2                      
/      x\    /          2    x\   /             x\     /          2    x\
\10 + e /*cos\-6*x + 5*x  + e / - \-6 + 10*x + e / *sin\-6*x + 5*x  + e /

$$\left(e^{x} + 10\right) \cos{\left(5 x^{2} - 6 x + e^{x} \right)} - \left(10 x + e^{x} - 6\right)^{2} \sin{\left(5 x^{2} - 6 x + e^{x} \right)}$$

Simplify

The third derivative [src]

                                           3                                                                           
   /          2    x\  x   /             x\     /          2    x\     /      x\ /             x\    /          2    x\
cos\-6*x + 5*x  + e /*e  - \-6 + 10*x + e / *cos\-6*x + 5*x  + e / - 3*\10 + e /*\-6 + 10*x + e /*sin\-6*x + 5*x  + e /

$$- 3 \left(e^{x} + 10\right) \left(10 x + e^{x} - 6\right) \sin{\left(5 x^{2} - 6 x + e^{x} \right)} - \left(10 x + e^{x} - 6\right)^{3} \cos{\left(5 x^{2} - 6 x + e^{x} \right)} + e^{x} \cos{\left(5 x^{2} - 6 x + e^{x} \right)}$$

Simplify

The graph

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Derivative of y=(sin(5x²-6x+e^x))

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The solution