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Derivative of x^5*cos(x)

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 5       
x *cos(x)

$$x^{5} \cos{\left(x \right)}$$

x^5*cos(x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x^{5}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; 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: $- x^{5} \sin{\left(x \right)} + 5 x^{4} \cos{\left(x \right)}$
Now simplify:

$x^{4} \left(- x \sin{\left(x \right)} + 5 \cos{\left(x \right)}\right)$

The answer is:

$x^{4} \left(- x \sin{\left(x \right)} + 5 \cos{\left(x \right)}\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   5             4       
- x *sin(x) + 5*x *cos(x)

$$- x^{5} \sin{\left(x \right)} + 5 x^{4} \cos{\left(x \right)}$$

Simplify

The second derivative [src]

 3 /             2                     \
x *\20*cos(x) - x *cos(x) - 10*x*sin(x)/

$$x^{3} \left(- x^{2} \cos{\left(x \right)} - 10 x \sin{\left(x \right)} + 20 \cos{\left(x \right)}\right)$$

Simplify

The third derivative [src]

 2 /             3                            2       \
x *\60*cos(x) + x *sin(x) - 60*x*sin(x) - 15*x *cos(x)/

$$x^{2} \left(x^{3} \sin{\left(x \right)} - 15 x^{2} \cos{\left(x \right)} - 60 x \sin{\left(x \right)} + 60 \cos{\left(x \right)}\right)$$

Simplify

The graph