Derivative of (tan(3x))/(tan(x))

The solution

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tan(3*x)
--------
 tan(x)

$$\frac{\tan{\left(3 x \right)}}{\tan{\left(x \right)}}$$

tan(3*x)/tan(x)

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /                                           /            2   \              /       2   \ /       2     \\
  |  /       2     \            /       2   \ |     1 + tan (x)|            3*\1 + tan (x)/*\1 + tan (3*x)/|
2*|9*\1 + tan (3*x)/*tan(3*x) + \1 + tan (x)/*|-1 + -----------|*tan(3*x) - -------------------------------|
  |                                           |          2     |                         tan(x)            |
  \                                           \       tan (x)  /                                           /
------------------------------------------------------------------------------------------------------------
                                                   tan(x)

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

Simplify

The third derivative [src]

  /                                                                                                                                                                                      /            2   \\
  |                                                                                                                                                        /       2   \ /       2     \ |     1 + tan (x)||
  |  /                               2                  3\                                                                                               9*\1 + tan (x)/*\1 + tan (3*x)/*|-1 + -----------||
  |  |                  /       2   \      /       2   \ |               /       2     \ /         2     \      /       2   \ /       2     \                                            |          2     ||
  |  |         2      5*\1 + tan (x)/    3*\1 + tan (x)/ |            27*\1 + tan (3*x)/*\1 + 3*tan (3*x)/   27*\1 + tan (x)/*\1 + tan (3*x)/*tan(3*x)                                   \       tan (x)  /|
2*|- |2 + 2*tan (x) - ---------------- + ----------------|*tan(3*x) + ------------------------------------ - ----------------------------------------- + --------------------------------------------------|
  |  |                       2                  4        |                           tan(x)                                      2                                             tan(x)                      |
  \  \                    tan (x)            tan (x)     /                                                                    tan (x)                                                                      /

$$2 \left(\frac{9 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(3 x \right)} + 1\right)}{\tan{\left(x \right)}} - \frac{27 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)}}{\tan^{2}{\left(x \right)}} + \frac{27 \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(3 \tan^{2}{\left(3 x \right)} + 1\right)}{\tan{\left(x \right)}} - \left(\frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{3}}{\tan^{4}{\left(x \right)}} - \frac{5 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + 2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(3 x \right)}\right)$$

Simplify

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Derivative of (tan(3x))/(tan(x))

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