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Integral of d{x}:
x^2*cos^2(x)
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Similar expressions
exp(x)*x^2*cos^2x

The derivative of the function/ x^2*cos^2(x)

Derivative of x^2*cos^2(x)

The solution

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 2    2   
x *cos (x)

$$x^{2} \cos^{2}{\left(x \right)}$$

x^2*cos(x)^2

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{2}$ goes to $2 x$
$g{\left(x \right)} = \cos^{2}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result of the chain rule is:
  
  $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
The result is: $- 2 x^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 x \cos^{2}{\left(x \right)}$
Now simplify:

$x \left(- x \sin{\left(2 x \right)} + \cos{\left(2 x \right)} + 1\right)$

The answer is:

$x \left(- x \sin{\left(2 x \right)} + \cos{\left(2 x \right)} + 1\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2         2              
2*x*cos (x) - 2*x *cos(x)*sin(x)

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

Simplify

The second derivative [src]

  /   2       2 /   2         2   \                    \
2*\cos (x) + x *\sin (x) - cos (x)/ - 4*x*cos(x)*sin(x)/

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of x^2*cos^2(x)

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The solution