Derivative of tgx

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tan(x)

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

tan(x)

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
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:

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

$\frac{1}{\cos^{2}{\left(x \right)}}$

The answer is:

$\frac{1}{\cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2   
1 + tan (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph