Mister Exam

Other calculators


sinx+(1/2)cotx

Derivative of sinx+(1/2)cotx

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

The graph:

from to

Piecewise:

The solution

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

    1. The derivative of sine is cosine:

    2. 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:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
         2            
  1   cot (x)         
- - - ------- + cos(x)
  2      2            
$$\cos{\left(x \right)} - \frac{\cot^{2}{\left(x \right)}}{2} - \frac{1}{2}$$
The second derivative [src]
          /       2   \       
-sin(x) + \1 + cot (x)/*cot(x)
$$\left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \sin{\left(x \right)}$$
The third derivative [src]
 /             2                                   \
 |/       2   \         2    /       2   \         |
-\\1 + cot (x)/  + 2*cot (x)*\1 + cot (x)/ + cos(x)/
$$- (\left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + \cos{\left(x \right)})$$
The graph
Derivative of sinx+(1/2)cotx