Mister Exam

Derivative of y=sin³x*cos²x

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   3       2   
sin (x)*cos (x)
$$\sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}$$
d /   3       2   \
--\sin (x)*cos (x)/
dx                 
$$\frac{d}{d x} \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}$$
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of sine is cosine:

      The result of the chain rule is:

    ; to find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of cosine is negative sine:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The first derivative [src]
       4                  3       2   
- 2*sin (x)*cos(x) + 3*cos (x)*sin (x)
$$- 2 \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}$$
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)
$$\left(- 3 \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} + 2 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 12 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}$$
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)
$$\left(18 \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} + 18 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 3 \cdot \left(7 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} + 8 \sin^{4}{\left(x \right)}\right) \cos{\left(x \right)}$$