Derivative of tan^-1(t)

The solution

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  1   
------
tan(t)

$$\frac{1}{\tan{\left(t \right)}}$$

1/tan(t)

Detail solution

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

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

$- \frac{1}{\sin^{2}{\left(t \right)}}$

The answer is:

$- \frac{1}{\sin^{2}{\left(t \right)}}$

The graph

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The first derivative [src]

        2   
-1 - tan (t)
------------
     2      
  tan (t)

$$\frac{- \tan^{2}{\left(t \right)} - 1}{\tan^{2}{\left(t \right)}}$$

Simplify

The second derivative [src]

                /            2   \
  /       2   \ |     1 + tan (t)|
2*\1 + tan (t)/*|-1 + -----------|
                |          2     |
                \       tan (t)  /
----------------------------------
              tan(t)

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of tan^-1(t)

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The solution