Mister Exam

Other calculators


y=√[tg(3x^2)]

Derivative of y=√[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 / 
$$\sqrt{\tan{\left(3 x^{2} \right)}}$$
  /   ___________\
d |  /    /   2\ |
--\\/  tan\3*x / /
dx                
$$\frac{d}{d x} \sqrt{\tan{\left(3 x^{2} \right)}}$$
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/   2\\
3*x*\1 + tan \3*x //
--------------------
      ___________   
     /    /   2\    
   \/  tan\3*x /    
$$\frac{3 x \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)}{\sqrt{\tan{\left(3 x^{2} \right)}}}$$
The second derivative [src]
                   /                          ___________      2 /       2/   2\\\
  /       2/   2\\ |      1              2   /    /   2\    3*x *\1 + tan \3*x //|
3*\1 + tan \3*x //*|-------------- + 12*x *\/  tan\3*x /  - ---------------------|
                   |   ___________                                  3/2/   2\    |
                   |  /    /   2\                                tan   \3*x /    |
                   \\/  tan\3*x /                                                /
$$3 \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right) \left(- \frac{3 x^{2} \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)}{\tan^{\frac{3}{2}}{\left(3 x^{2} \right)}} + 12 x^{2} \sqrt{\tan{\left(3 x^{2} \right)}} + \frac{1}{\sqrt{\tan{\left(3 x^{2} \right)}}}\right)$$
The third derivative [src]
                      /                                                                                                      2\
                      |     ___________          2/   2\                           2 /       2/   2\\      2 /       2/   2\\ |
     /       2/   2\\ |    /    /   2\    1 + tan \3*x /       2    3/2/   2\   4*x *\1 + tan \3*x //   3*x *\1 + tan \3*x // |
27*x*\1 + tan \3*x //*|4*\/  tan\3*x /  - -------------- + 16*x *tan   \3*x / - --------------------- + ----------------------|
                      |                       3/2/   2\                                ___________              5/2/   2\     |
                      |                    tan   \3*x /                               /    /   2\            tan   \3*x /     |
                      \                                                             \/  tan\3*x /                             /
$$27 x \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right) \left(\frac{3 x^{2} \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)^{2}}{\tan^{\frac{5}{2}}{\left(3 x^{2} \right)}} - \frac{4 x^{2} \left(\tan^{2}{\left(3 x^{2} \right)} + 1\right)}{\sqrt{\tan{\left(3 x^{2} \right)}}} + 16 x^{2} \tan^{\frac{3}{2}}{\left(3 x^{2} \right)} - \frac{\tan^{2}{\left(3 x^{2} \right)} + 1}{\tan^{\frac{3}{2}}{\left(3 x^{2} \right)}} + 4 \sqrt{\tan{\left(3 x^{2} \right)}}\right)$$
The graph
Derivative of y=√[tg(3x^2)]