Derivative of y=x⁴cosx

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 4       
x *cos(x)

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

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{4}$ ; 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: $- x^{4} \sin{\left(x \right)} + 4 x^{3} \cos{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph