Mister Exam

Derivative of y=ctg(x)-sqrt(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
           ___
cot(x) - \/ x 
$$- \sqrt{x} + \cot{\left(x \right)}$$
d /           ___\
--\cot(x) - \/ x /
dx                
$$\frac{d}{d x} \left(- \sqrt{x} + \cot{\left(x \right)}\right)$$
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         1   
-1 - cot (x) - -------
                   ___
               2*\/ x 
$$- \cot^{2}{\left(x \right)} - 1 - \frac{1}{2 \sqrt{x}}$$
The second derivative [src]
  1        /       2   \       
------ + 2*\1 + cot (x)/*cot(x)
   3/2                         
4*x                            
$$2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{1}{4 x^{\frac{3}{2}}}$$
The third derivative [src]
 /               2                                   \
 |  /       2   \      3           2    /       2   \|
-|2*\1 + cot (x)/  + ------ + 4*cot (x)*\1 + cot (x)/|
 |                      5/2                          |
 \                   8*x                             /
$$- (2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + \frac{3}{8 x^{\frac{5}{2}}})$$
The graph
Derivative of y=ctg(x)-sqrt(x)