Derivative of y=sin³x*cos²x

The solution

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   3       2   
sin (x)*cos (x)

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

d /   3       2   \
--\sin (x)*cos (x)/
dx

$$\frac{d}{d x} \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin^{3}{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
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:
  
  $3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
$g{\left(x \right)} = \cos^{2}{\left(x \right)}$ ; 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: $- 2 \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}$
Now simplify:

$\left(3 - 5 \sin^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)}$

The answer is:

$\left(3 - 5 \sin^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)}$

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/     4           2    /       2           2   \         2    /   2         2   \         2    /   2           2   \\       
\8*sin (x) - 3*cos (x)*\- 2*cos (x) + 7*sin (x)/ + 18*sin (x)*\sin (x) - cos (x)/ + 18*sin (x)*\sin (x) - 2*cos (x)//*cos(x)

$$\left(18 \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} + 18 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 3 \cdot \left(7 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} + 8 \sin^{4}{\left(x \right)}\right) \cos{\left(x \right)}$$

Simplify

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Derivative of y=sin³x*cos²x

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