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The derivative of the function/ y=tgx/(x-2)^2

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

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

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \tan{\left(x \right)}$ and $g{\left(x \right)} = \left(x - 2\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x - 2$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)$ :
  1. Differentiate $x - 2$ term by term:
    1. The derivative of the constant $-2$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $2 x - 4$
Now plug in to the quotient rule:

$\frac{\frac{\left(x - 2\right)^{2} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \left(2 x - 4\right) \tan{\left(x \right)}}{\left(x - 2\right)^{4}}$
Now simplify:

$\frac{x - \sin{\left(2 x \right)} - 2}{\left(x - 2\right)^{3} \cos^{2}{\left(x \right)}}$

The answer is:

$\frac{x - \sin{\left(2 x \right)} - 2}{\left(x - 2\right)^{3} \cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2                      
1 + tan (x)   (4 - 2*x)*tan(x)
----------- + ----------------
         2               4    
  (x - 2)         (x - 2)

$$\frac{\left(4 - 2 x\right) \tan{\left(x \right)}}{\left(x - 2\right)^{4}} + \frac{\tan^{2}{\left(x \right)} + 1}{\left(x - 2\right)^{2}}$$

Simplify

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)

$$\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x - 2} + \frac{3 \tan{\left(x \right)}}{\left(x - 2\right)^{2}}\right)}{\left(x - 2\right)^{2}}$$

Simplify

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)

$$\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x - 2} + \frac{9 \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(x - 2\right)^{2}} - \frac{12 \tan{\left(x \right)}}{\left(x - 2\right)^{3}}\right)}{\left(x - 2\right)^{2}}$$

Simplify

The graph

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

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Piecewise:

The solution