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y=tgx/(x-2)^2

Derivative of y=tgx/(x-2)^2

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
 tan(x) 
--------
       2
(x - 2) 
$$\frac{\tan{\left(x \right)}}{\left(x - 2\right)^{2}}$$
tan(x)/(x - 2)^2
Detail solution
  1. Apply the quotient rule, which is:

    and .

    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. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Differentiate term by term:

        1. The derivative of the constant is zero.

        2. Apply the power rule: goes to

        The result is:

      The result of the chain rule is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
       2                      
1 + tan (x)   (4 - 2*x)*tan(x)
----------- + ----------------
         2               4    
  (x - 2)         (x - 2)     
$$\frac{\left(4 - 2 x\right) \tan{\left(x \right)}}{\left(x - 2\right)^{4}} + \frac{\tan^{2}{\left(x \right)} + 1}{\left(x - 2\right)^{2}}$$
The second derivative [src]
  /                         /       2   \            \
  |/       2   \          2*\1 + tan (x)/    3*tan(x)|
2*|\1 + tan (x)/*tan(x) - --------------- + ---------|
  |                            -2 + x               2|
  \                                         (-2 + x) /
------------------------------------------------------
                              2                       
                      (-2 + x)                        
$$\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x - 2} + \frac{3 \tan{\left(x \right)}}{\left(x - 2\right)^{2}}\right)}{\left(x - 2\right)^{2}}$$
The third derivative [src]
  /                                              /       2   \     /       2   \       \
  |/       2   \ /         2   \   12*tan(x)   9*\1 + tan (x)/   6*\1 + tan (x)/*tan(x)|
2*|\1 + tan (x)/*\1 + 3*tan (x)/ - --------- + --------------- - ----------------------|
  |                                        3              2              -2 + x        |
  \                                (-2 + x)       (-2 + x)                             /
----------------------------------------------------------------------------------------
                                               2                                        
                                       (-2 + x)                                         
$$\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x - 2} + \frac{9 \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(x - 2\right)^{2}} - \frac{12 \tan{\left(x \right)}}{\left(x - 2\right)^{3}}\right)}{\left(x - 2\right)^{2}}$$
The graph
Derivative of y=tgx/(x-2)^2