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Similar expressions
tan(2*x)/(x-1)

The derivative of the function/ tan(2*x)/(x+1)

Derivative of tan(2*x)/(x+1)

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \tan{\left(2 x \right)}$ and $g{\left(x \right)} = x + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
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:
  
  $\frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

$\frac{\frac{\left(x + 1\right) \left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right)}{\cos^{2}{\left(2 x \right)}} - \tan{\left(2 x \right)}}{\left(x + 1\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                 /       2     \                                            /       2     \         \
  |  3*tan(2*x)   6*\1 + tan (2*x)/     /       2     \ /         2     \   12*\1 + tan (2*x)/*tan(2*x)|
2*|- ---------- + ----------------- + 8*\1 + tan (2*x)/*\1 + 3*tan (2*x)/ - ---------------------------|
  |          3                2                                                        1 + x           |
  \   (1 + x)          (1 + x)                                                                         /
--------------------------------------------------------------------------------------------------------
                                                 1 + x

$$\frac{2 \left(8 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(3 \tan^{2}{\left(2 x \right)} + 1\right) - \frac{12 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)}}{x + 1} + \frac{6 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{\left(x + 1\right)^{2}} - \frac{3 \tan{\left(2 x \right)}}{\left(x + 1\right)^{3}}\right)}{x + 1}$$

Simplify

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Derivative of tan(2*x)/(x+1)

The graph:

Piecewise:

The solution