Mister Exam

Derivative of y=sin^3x+sin3x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   3              
sin (x) + sin(3*x)
$$\sin^{3}{\left(x \right)} + \sin{\left(3 x \right)}$$
d /   3              \
--\sin (x) + sin(3*x)/
dx                    
$$\frac{d}{d x} \left(\sin^{3}{\left(x \right)} + \sin{\left(3 x \right)}\right)$$
Detail solution
  1. Differentiate term by term:

    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:

    4. Let .

    5. The derivative of sine is cosine:

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

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
                  2          
3*cos(3*x) + 3*sin (x)*cos(x)
$$3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 3 \cos{\left(3 x \right)}$$
The second derivative [src]
  /     3                        2          \
3*\- sin (x) - 3*sin(3*x) + 2*cos (x)*sin(x)/
$$3 \left(- \sin^{3}{\left(x \right)} + 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)} - 3 \sin{\left(3 x \right)}\right)$$
The third derivative [src]
  /                   3           2          \
3*\-9*cos(3*x) + 2*cos (x) - 7*sin (x)*cos(x)/
$$3 \left(- 7 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 2 \cos^{3}{\left(x \right)} - 9 \cos{\left(3 x \right)}\right)$$
The graph
Derivative of y=sin^3x+sin3x