Derivative of y=sinxcosx²

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          2   
sin(x)*cos (x)

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

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

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

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
$g{\left(x \right)} = \cos^{2}{\left(x \right)}$ ; 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: $- 2 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos^{3}{\left(x \right)}$
Now simplify:

$\left(1 - 3 \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}$

The answer is:

$\left(1 - 3 \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/       2           2   \       
\- 7*cos (x) + 2*sin (x)/*sin(x)

$$\left(2 \sin^{2}{\left(x \right)} - 7 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}$$

Simplify

The third derivative [src]

/       2            2   \       
\- 7*cos (x) + 20*sin (x)/*cos(x)

$$\left(20 \sin^{2}{\left(x \right)} - 7 \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}$$

Simplify

The graph