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Derivative of x^4*sin(x)

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 4       
x *sin(x)

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

x^4*sin(x)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph