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y=4•ctgx+12x^2

Derivative of y=4•ctgx+12x^2

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

The graph:

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Piecewise:

The solution

You have entered [src]
               2
4*cot(x) + 12*x 
$$12 x^{2} + 4 \cot{\left(x \right)}$$
d /               2\
--\4*cot(x) + 12*x /
dx                  
$$\frac{d}{d x} \left(12 x^{2} + 4 \cot{\left(x \right)}\right)$$
Detail solution
  1. Differentiate term by term:

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

      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:

      So, the result is:

    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          
-4 - 4*cot (x) + 24*x
$$- 4 \cot^{2}{\left(x \right)} + 24 x - 4$$
The second derivative [src]
  /    /       2   \       \
8*\3 + \1 + cot (x)/*cot(x)/
$$8 \left(\left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + 3\right)$$
The third derivative [src]
   /       2   \ /         2   \
-8*\1 + cot (x)/*\1 + 3*cot (x)/
$$- 8 \left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right)$$
The graph
Derivative of y=4•ctgx+12x^2