Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 9/x
Derivative of x^-5
Derivative of x^-3
Derivative of x^2*e^x
Identical expressions
y=tg^ two (x- two)/tg(x+ three)
y equally tg squared (x minus 2) divide by tg(x plus 3)
y equally tg to the power of two (x minus two) divide by tg(x plus three)
y=tg2(x-2)/tg(x+3)
y=tg2x-2/tgx+3
y=tg²(x-2)/tg(x+3)
y=tg to the power of 2(x-2)/tg(x+3)
y=tg^2x-2/tgx+3
y=tg^2(x-2) divide by tg(x+3)
Similar expressions
y=tg^2(x+2)/tg(x+3)
y=tg^2(x-2)/tg(x-3)

The derivative of the function/ y=tg^2(x-2)/tg(x+3)

Derivative of y=tg^2(x-2)/tg(x+3)

The solution

You have entered [src]

   2       
tan (x - 2)
-----------
 tan(x + 3)

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

tan(x - 2)^2/tan(x + 3)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \tan^{2}{\left(x - 2 \right)}$ and $g{\left(x \right)} = \tan{\left(x + 3 \right)}$ .

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

$\frac{\frac{2 \left(\sin^{2}{\left(x - 2 \right)} + \cos^{2}{\left(x - 2 \right)}\right) \tan{\left(x - 2 \right)} \tan{\left(x + 3 \right)}}{\cos^{2}{\left(x - 2 \right)}} - \frac{\left(\sin^{2}{\left(x + 3 \right)} + \cos^{2}{\left(x + 3 \right)}\right) \tan^{2}{\left(x - 2 \right)}}{\cos^{2}{\left(x + 3 \right)}}}{\tan^{2}{\left(x + 3 \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2        /        2       \   /         2       \           
tan (x - 2)*\-1 - tan (x + 3)/   \2 + 2*tan (x - 2)/*tan(x - 2)
------------------------------ + ------------------------------
            2                              tan(x + 3)          
         tan (x + 3)

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

Simplify

The second derivative [src]

  /                                                                         /            2       \     /       2        \ /       2       \            \
  |/       2        \ /         2        \      2         /       2       \ |     1 + tan (3 + x)|   2*\1 + tan (-2 + x)/*\1 + tan (3 + x)/*tan(-2 + x)|
2*|\1 + tan (-2 + x)/*\1 + 3*tan (-2 + x)/ + tan (-2 + x)*\1 + tan (3 + x)/*|-1 + ---------------| - --------------------------------------------------|
  |                                                                         |          2         |                       tan(3 + x)                    |
  \                                                                         \       tan (3 + x)  /                                                     /
--------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                       tan(3 + x)

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

Simplify

The third derivative [src]

  /                                                                                                                                                                                                                                                /            2       \            \
  |                                                                                                                                                                                                           /       2        \ /       2       \ |     1 + tan (3 + x)|            |
  |               /                                       2                      3\                                                                                                                         6*\1 + tan (-2 + x)/*\1 + tan (3 + x)/*|-1 + ---------------|*tan(-2 + x)|
  |               |                      /       2       \      /       2       \ |     /       2        \ /       2       \ /         2        \     /       2        \ /         2        \                                                      |          2         |            |
  |     2         |         2          5*\1 + tan (3 + x)/    3*\1 + tan (3 + x)/ |   3*\1 + tan (-2 + x)/*\1 + tan (3 + x)/*\1 + 3*tan (-2 + x)/   4*\1 + tan (-2 + x)/*\2 + 3*tan (-2 + x)/*tan(-2 + x)                                          \       tan (3 + x)  /            |
2*|- tan (-2 + x)*|2 + 2*tan (3 + x) - -------------------- + --------------------| - ----------------------------------------------------------- + ----------------------------------------------------- + -------------------------------------------------------------------------|
  |               |                           2                      4            |                              2                                                        tan(3 + x)                                                        tan(3 + x)                               |
  \               \                        tan (3 + x)            tan (3 + x)     /                           tan (3 + x)                                                                                                                                                            /

$$2 \left(\frac{6 \left(\frac{\tan^{2}{\left(x + 3 \right)} + 1}{\tan^{2}{\left(x + 3 \right)}} - 1\right) \left(\tan^{2}{\left(x - 2 \right)} + 1\right) \left(\tan^{2}{\left(x + 3 \right)} + 1\right) \tan{\left(x - 2 \right)}}{\tan{\left(x + 3 \right)}} - \frac{3 \left(\tan^{2}{\left(x - 2 \right)} + 1\right) \left(3 \tan^{2}{\left(x - 2 \right)} + 1\right) \left(\tan^{2}{\left(x + 3 \right)} + 1\right)}{\tan^{2}{\left(x + 3 \right)}} + \frac{4 \left(\tan^{2}{\left(x - 2 \right)} + 1\right) \left(3 \tan^{2}{\left(x - 2 \right)} + 2\right) \tan{\left(x - 2 \right)}}{\tan{\left(x + 3 \right)}} - \left(\frac{3 \left(\tan^{2}{\left(x + 3 \right)} + 1\right)^{3}}{\tan^{4}{\left(x + 3 \right)}} - \frac{5 \left(\tan^{2}{\left(x + 3 \right)} + 1\right)^{2}}{\tan^{2}{\left(x + 3 \right)}} + 2 \tan^{2}{\left(x + 3 \right)} + 2\right) \tan^{2}{\left(x - 2 \right)}\right)$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=tg^2(x-2)/tg(x+3)

The graph:

Piecewise:

The solution