Derivative of x^-2+6tanx

The solution

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1            
-- + 6*tan(x)
 2           
x

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

x^(-2) + 6*tan(x)

Detail solution

Differentiate $6 \tan{\left(x \right)} + \frac{1}{x^{2}}$ term by term:
1. Apply the power rule: $\frac{1}{x^{2}}$ goes to $- \frac{2}{x^{3}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  So, the result is: $\frac{6 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$
The result is: $\frac{6 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \frac{2}{x^{3}}$
Now simplify:

$\frac{6}{\cos^{2}{\left(x \right)}} - \frac{2}{x^{3}}$

The answer is:

$\frac{6}{\cos^{2}{\left(x \right)}} - \frac{2}{x^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    2         2   
6 - -- + 6*tan (x)
     3            
    x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /             2                               \
   |/       2   \    2         2    /       2   \|
12*|\1 + tan (x)/  - -- + 2*tan (x)*\1 + tan (x)/|
   |                  5                          |
   \                 x                           /

$$12 \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - \frac{2}{x^{5}}\right)$$

Simplify

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Derivative of x^-2+6tanx

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