Derivative of sin^2xcos^2x

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   2       2   
sin (x)*cos (x)

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

d /   2       2   \
--\sin (x)*cos (x)/
dx

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

Transform

Detail solution

Apply the product rule:

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

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

$\frac{\sin{\left(4 x \right)}}{2}$

The answer is:

$\frac{\sin{\left(4 x \right)}}{2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       3                  3          
- 2*sin (x)*cos(x) + 2*cos (x)*sin(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       2           2   \              
8*\- 4*cos (x) + 4*sin (x)/*cos(x)*sin(x)

$$8 \cdot \left(4 \sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The graph

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Derivative of sin^2xcos^2x

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The solution