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y=tg^2x-ctg^2x

Derivative of y=tg^2x-ctg^2x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   2         2   
tan (x) - cot (x)
$$\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}$$
d /   2         2   \
--\tan (x) - cot (x)/
dx                   
$$\frac{d}{d x} \left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)$$
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. Rewrite the function to be differentiated:

      2. Apply the quotient rule, which is:

        and .

        To find :

        1. The derivative of sine is cosine:

        To find :

        1. The derivative of cosine is negative sine:

        Now plug in to the quotient rule:

      The result of the chain rule is:

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

      1. Let .

      2. Apply the power rule: goes to

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

        1. There are multiple ways to do this derivative.

          Method #1

          1. Rewrite the function to be differentiated:

          2. Let .

          3. Apply the power rule: goes to

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

            The result of the chain rule is:

          Method #2

          1. Rewrite the function to be differentiated:

          2. Apply the quotient rule, which is:

            and .

            To find :

            1. The derivative of cosine is negative sine:

            To find :

            1. The derivative of sine is cosine:

            Now plug in to the quotient rule:

        The result of the chain rule is:

      So, the result is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
/         2   \          /          2   \       
\2 + 2*tan (x)/*tan(x) - \-2 - 2*cot (x)/*cot(x)
$$\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} - \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}$$
The second derivative [src]
  /             2                2                                                    \
  |/       2   \    /       2   \         2    /       2   \        2    /       2   \|
2*\\1 + tan (x)/  - \1 + cot (x)/  - 2*cot (x)*\1 + cot (x)/ + 2*tan (x)*\1 + tan (x)//
$$2 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2} - \left(\cot^{2}{\left(x \right)} + 1\right)^{2}\right)$$
The third derivative [src]
  /                                                               2                         2       \
  |   3    /       2   \      3    /       2   \     /       2   \             /       2   \        |
8*\cot (x)*\1 + cot (x)/ + tan (x)*\1 + tan (x)/ + 2*\1 + cot (x)/ *cot(x) + 2*\1 + tan (x)/ *tan(x)/
$$8 \cdot \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan^{3}{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{3}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \cot{\left(x \right)}\right)$$
The graph
Derivative of y=tg^2x-ctg^2x