Mister Exam

Derivative of ctg5x+lnx

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
cot(5*x) + log(x)
$$\log{\left(x \right)} + \cot{\left(5 x \right)}$$
cot(5*x) + log(x)
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. 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:

      Method #2

      1. Rewrite the function to be differentiated:

      2. Apply the quotient rule, which is:

        and .

        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:

        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:

        Now plug in to the quotient rule:

    2. The derivative of is .

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
     1        2     
-5 + - - 5*cot (5*x)
     x              
$$- 5 \cot^{2}{\left(5 x \right)} - 5 + \frac{1}{x}$$
The second derivative [src]
  1       /       2     \         
- -- + 50*\1 + cot (5*x)/*cot(5*x)
   2                              
  x                               
$$50 \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)} - \frac{1}{x^{2}}$$
The third derivative [src]
  /                        2                                \
  |1        /       2     \           2      /       2     \|
2*|-- - 125*\1 + cot (5*x)/  - 250*cot (5*x)*\1 + cot (5*x)/|
  | 3                                                       |
  \x                                                        /
$$2 \left(- 125 \left(\cot^{2}{\left(5 x \right)} + 1\right)^{2} - 250 \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot^{2}{\left(5 x \right)} + \frac{1}{x^{3}}\right)$$