Mister Exam

Derivative of tg(3x)^2

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   2     
tan (3*x)
$$\tan^{2}{\left(3 x \right)}$$
d /   2     \
--\tan (3*x)/
dx           
$$\frac{d}{d x} \tan^{2}{\left(3 x \right)}$$
Detail solution
  1. Let .

  2. Apply the power rule: goes to

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

    1. Rewrite the function to be differentiated:

    2. Apply the quotient rule, which is:

      and .

      To find :

      1. Let .

      2. The derivative of sine is cosine:

      3. 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:

      To find :

      1. Let .

      2. The derivative of cosine is negative sine:

      3. 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:

      Now plug in to the quotient rule:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
/         2     \         
\6 + 6*tan (3*x)/*tan(3*x)
$$\left(6 \tan^{2}{\left(3 x \right)} + 6\right) \tan{\left(3 x \right)}$$
The second derivative [src]
   /       2     \ /         2     \
18*\1 + tan (3*x)/*\1 + 3*tan (3*x)/
$$18 \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(3 \tan^{2}{\left(3 x \right)} + 1\right)$$
The third derivative [src]
    /       2     \ /         2     \         
216*\1 + tan (3*x)/*\2 + 3*tan (3*x)/*tan(3*x)
$$216 \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(3 \tan^{2}{\left(3 x \right)} + 2\right) \tan{\left(3 x \right)}$$
3-th derivative [src]
    /       2     \ /         2     \         
216*\1 + tan (3*x)/*\2 + 3*tan (3*x)/*tan(3*x)
$$216 \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(3 \tan^{2}{\left(3 x \right)} + 2\right) \tan{\left(3 x \right)}$$
The graph
Derivative of tg(3x)^2