Mister Exam

Lang:

Derivative of x^6*cos(x)

You have entered [src]

 6       
x *cos(x)

$$x^{6} \cos{\left(x \right)}$$

x^6*cos(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{6}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{6}$ goes to $6 x^{5}$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; 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: $- x^{6} \sin{\left(x \right)} + 6 x^{5} \cos{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   6             5       
- x *sin(x) + 6*x *cos(x)

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

Simplify

The second derivative [src]

 4 /             2                     \
x *\30*cos(x) - x *cos(x) - 12*x*sin(x)/

$$x^{4} \left(- x^{2} \cos{\left(x \right)} - 12 x \sin{\left(x \right)} + 30 \cos{\left(x \right)}\right)$$

Simplify

The third derivative [src]

 3 /              3                            2       \
x *\120*cos(x) + x *sin(x) - 90*x*sin(x) - 18*x *cos(x)/

$$x^{3} \left(x^{3} \sin{\left(x \right)} - 18 x^{2} \cos{\left(x \right)} - 90 x \sin{\left(x \right)} + 120 \cos{\left(x \right)}\right)$$

Simplify

The graph