Mister Exam

Derivative of y=ctgx-8x^3

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
            3
cot(x) - 8*x 
$$- 8 x^{3} + \cot{\left(x \right)}$$
Detail solution
  1. Differentiate term by term:

    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 :

        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:

      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:

    2. 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 is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
        2          2
-1 - cot (x) - 24*x 
$$- 24 x^{2} - \cot^{2}{\left(x \right)} - 1$$
The second derivative [src]
  /        /       2   \       \
2*\-24*x + \1 + cot (x)/*cot(x)/
$$2 \left(- 24 x + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right)$$
The third derivative [src]
   /                  2                          \
   |     /       2   \         2    /       2   \|
-2*\24 + \1 + cot (x)/  + 2*cot (x)*\1 + cot (x)//
$$- 2 \left(\left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + 24\right)$$
The graph
Derivative of y=ctgx-8x^3