Derivative of tgx+tg^3x

The solution

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            3   
tan(x) + tan (x)

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

d /            3   \
--\tan(x) + tan (x)/
dx

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

Transform

Detail solution

Differentiate $\tan^{3}{\left(x \right)} + \tan{\left(x \right)}$ term by term:
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:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
3. Let $u = \tan{\left(x \right)}$ .
4. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
5. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
  1. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
  The result of the chain rule is:
  
  $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result is: $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                /                   2                                                   \
  /       2   \ |      /       2   \         2           4            2    /       2   \|
2*\1 + tan (x)/*\1 + 3*\1 + tan (x)/  + 3*tan (x) + 6*tan (x) + 21*tan (x)*\1 + tan (x)//

$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(6 \tan^{4}{\left(x \right)} + 21 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 3 \tan^{2}{\left(x \right)} + 1\right)$$

Simplify

The graph

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Derivative of tgx+tg^3x

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The solution