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Derivative of:
Derivative of 2^(3*x)
Derivative of 3*x^2
Derivative of x^(1/5)
Derivative of sin(x)*cos(x)
Identical expressions
tan(x)- one / three *tan(x)^(three)+ one / five *tan(x)^(five)
tangent of (x) minus 1 divide by 3 multiply by tangent of (x) to the power of (3) plus 1 divide by 5 multiply by tangent of (x) to the power of (5)
tangent of (x) minus one divide by three multiply by tangent of (x) to the power of (three) plus one divide by five multiply by tangent of (x) to the power of (five)
tan(x)-1/3*tan(x)(3)+1/5*tan(x)(5)
tanx-1/3*tanx3+1/5*tanx5
tan(x)-1/3tan(x)^(3)+1/5tan(x)^(5)
tan(x)-1/3tan(x)(3)+1/5tan(x)(5)
tanx-1/3tanx3+1/5tanx5
tanx-1/3tanx^3+1/5tanx^5
tan(x)-1 divide by 3*tan(x)^(3)+1 divide by 5*tan(x)^(5)
Similar expressions
tan(x)-1/3*tan(x)^(3)-1/5*tan(x)^(5)
tan(x)+1/3*tan(x)^(3)+1/5*tan(x)^(5)

The derivative of the function/ tan(x)-1/3*tan(x)^(3)+1/5*tan(x)^(5)

Derivative of tan(x)-1/3tan(x)^(3)+1/5tan(x)^(5)

The solution

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            3         5   
         tan (x)   tan (x)
tan(x) - ------- + -------
            3         5

$$\frac{\tan^{5}{\left(x \right)}}{5} - \frac{\tan^{3}{\left(x \right)}}{3} + \tan{\left(x \right)}$$

  /            3         5   \
d |         tan (x)   tan (x)|
--|tan(x) - ------- + -------|
dx\            3         5   /

$$\frac{d}{d x} \left(\frac{\tan^{5}{\left(x \right)}}{5} - \frac{\tan^{3}{\left(x \right)}}{3} + \tan{\left(x \right)}\right)$$

Transform

Detail solution

Differentiate $\frac{\tan^{5}{\left(x \right)}}{5} - \frac{\tan^{3}{\left(x \right)}}{3} + \tan{\left(x \right)}$ 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. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \tan{\left(x \right)}$ .
    2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
    3. 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:
      
      $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    So, the result is: $\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  So, the result is: $- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \tan{\left(x \right)}$ .
  2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$
  3. 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:
    
    $\frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  So, the result is: $\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result is: $\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                /                 2                                                                                2                                   \
  /       2   \ |    /       2   \         4           6           2           2    /       2   \     /       2   \     2            4    /       2   \|
2*\1 + tan (x)/*\1 - \1 + tan (x)/  - 2*tan (x) + 2*tan (x) + 3*tan (x) - 7*tan (x)*\1 + tan (x)/ + 6*\1 + tan (x)/ *tan (x) + 13*tan (x)*\1 + tan (x)//

$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)} - \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 13 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{4}{\left(x \right)} - 7 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \tan^{6}{\left(x \right)} - 2 \tan^{4}{\left(x \right)} + 3 \tan^{2}{\left(x \right)} + 1\right)$$

Simplify

The graph

Derivative of tan(x)-1/3*tan(x)^(3)+1/5*tan(x)^(5)

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Identical expressions

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Derivative of tan(x)-1/3tan(x)^(3)+1/5tan(x)^(5)

The graph:

Piecewise:

The solution

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Identical expressions

Similar expressions

Derivative of tan(x)-1/3*tan(x)^(3)+1/5*tan(x)^(5)

The graph:

Piecewise:

The solution

Derivative of tan(x)-1/3tan(x)^(3)+1/5tan(x)^(5)