Mister Exam

Derivative of y=sqrt(tanx)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
  ________
\/ tan(x) 
$$\sqrt{\tan{\left(x \right)}}$$
sqrt(tan(x))
Detail solution
  1. Let .

  2. Apply the power rule: goes to

  3. 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:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
       2   
1   tan (x)
- + -------
2      2   
-----------
   ________
 \/ tan(x) 
$$\frac{\frac{\tan^{2}{\left(x \right)}}{2} + \frac{1}{2}}{\sqrt{\tan{\left(x \right)}}}$$
The second derivative [src]
/       2   \ /                      2   \
|1   tan (x)| |    ________   1 + tan (x)|
|- + -------|*|4*\/ tan(x)  - -----------|
\4      4   / |                   3/2    |
              \                tan   (x) /
$$\left(- \frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} + 4 \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)}}{4} + \frac{1}{4}\right)$$
The third derivative [src]
              /                                                2\
/       2   \ |                 /       2   \     /       2   \ |
|1   tan (x)| |      3/2      4*\1 + tan (x)/   3*\1 + tan (x)/ |
|- + -------|*|16*tan   (x) - --------------- + ----------------|
\8      8   / |                    ________           5/2       |
              \                  \/ tan(x)         tan   (x)    /
$$\left(\frac{\tan^{2}{\left(x \right)}}{8} + \frac{1}{8}\right) \left(\frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{\frac{5}{2}}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{\sqrt{\tan{\left(x \right)}}} + 16 \tan^{\frac{3}{2}}{\left(x \right)}\right)$$
The graph
Derivative of y=sqrt(tanx)