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The derivative of the function/ tan(2*x)+4/(x-2)

Derivative of tan(2*x)+4/(x-2)

The solution

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             4  
tan(2*x) + -----
           x - 2

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

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

Detail solution

Differentiate $\tan{\left(2 x \right)} + \frac{4}{x - 2}$ term by term:
1. Rewrite the function to be differentiated:
  
  $\tan{\left(2 x \right)} = \frac{\sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(2 x \right)}$ and $g{\left(x \right)} = \cos{\left(2 x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 2 x$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result of the chain rule is:
    
    $2 \cos{\left(2 x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 2 x$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result of the chain rule is:
    
    $- 2 \sin{\left(2 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}$
3. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = x - 2$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)$ :
    1. Differentiate $x - 2$ term by term:
      1. Apply the power rule: $x$ goes to $1$
      2. The derivative of the constant $-2$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $- \frac{1}{\left(x - 2\right)^{2}}$
  So, the result is: $- \frac{4}{\left(x - 2\right)^{2}}$
The result is: $\frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} - \frac{4}{\left(x - 2\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

       4            2     
2 - -------- + 2*tan (2*x)
           2              
    (x - 2)

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

Simplify

The second derivative [src]

  /    1       /       2     \         \
8*|--------- + \1 + tan (2*x)/*tan(2*x)|
  |        3                           |
  \(-2 + x)                            /

$$8 \left(\left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} + \frac{1}{\left(x - 2\right)^{3}}\right)$$

Simplify

The third derivative [src]

  /                               2                              \
  |      3         /       2     \         2      /       2     \|
8*|- --------- + 2*\1 + tan (2*x)/  + 4*tan (2*x)*\1 + tan (2*x)/|
  |          4                                                   |
  \  (-2 + x)                                                    /

$$8 \left(2 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan^{2}{\left(2 x \right)} - \frac{3}{\left(x - 2\right)^{4}}\right)$$

Simplify

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Derivative of tan(2*x)+4/(x-2)

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The solution