Mister Exam

Derivative of y=tgx×lnx

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
tan(x)*log(x)
$$\log{\left(x \right)} \tan{\left(x \right)}$$
d                
--(tan(x)*log(x))
dx               
$$\frac{d}{d x} \log{\left(x \right)} \tan{\left(x \right)}$$
Detail solution
  1. Apply the product rule:

    ; to find :

    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:

    ; to find :

    1. The derivative of is .

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
tan(x)   /       2   \       
------ + \1 + tan (x)/*log(x)
  x                          
$$\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}$$
The second derivative [src]
             /       2   \                                
  tan(x)   2*\1 + tan (x)/     /       2   \              
- ------ + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)
     2            x                                       
    x                                                     
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{\tan{\left(x \right)}}{x^{2}}$$
The third derivative [src]
    /       2   \                                                         /       2   \       
  3*\1 + tan (x)/   2*tan(x)     /       2   \ /         2   \          6*\1 + tan (x)/*tan(x)
- --------------- + -------- + 2*\1 + tan (x)/*\1 + 3*tan (x)/*log(x) + ----------------------
          2             3                                                         x           
         x             x                                                                      
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2}} + \frac{2 \tan{\left(x \right)}}{x^{3}}$$
The graph
Derivative of y=tgx×lnx