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Derivative of cosx*(4x^4)

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          4
cos(x)*4*x

$$\cos{\left(x \right)} 4 x^{4}$$

d /          4\
--\cos(x)*4*x /
dx

$$\frac{d}{d x} \cos{\left(x \right)} 4 x^{4}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{4}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
  $g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result is: $- x^{4} \sin{\left(x \right)} + 4 x^{3} \cos{\left(x \right)}$
So, the result is: $- 4 x^{4} \sin{\left(x \right)} + 16 x^{3} \cos{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     4              3       
- 4*x *sin(x) + 16*x *cos(x)

$$- 4 x^{4} \sin{\left(x \right)} + 16 x^{3} \cos{\left(x \right)}$$

Simplify

The second derivative [src]

   2 /             2                    \
4*x *\12*cos(x) - x *cos(x) - 8*x*sin(x)/

$$4 x^{2} \left(- x^{2} \cos{\left(x \right)} - 8 x \sin{\left(x \right)} + 12 \cos{\left(x \right)}\right)$$

Simplify

The third derivative [src]

    /             3                            2       \
4*x*\24*cos(x) + x *sin(x) - 36*x*sin(x) - 12*x *cos(x)/

$$4 x \left(x^{3} \sin{\left(x \right)} - 12 x^{2} \cos{\left(x \right)} - 36 x \sin{\left(x \right)} + 24 \cos{\left(x \right)}\right)$$

Simplify

The graph