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Derivative of:
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Derivative of 1/tan(x)
Identical expressions
((x- one)^(two)×(x+ five))/(x+ one)^(three)
((x minus 1) to the power of (2)×(x plus 5)) divide by (x plus 1) to the power of (3)
((x minus one) to the power of (two)×(x plus five)) divide by (x plus one) to the power of (three)
((x-1)(2)×(x+5))/(x+1)(3)
x-12×x+5/x+13
x-1^2×x+5/x+1^3
((x-1)^(2)×(x+5)) divide by (x+1)^(3)
Similar expressions
((x+1)^(2)×(x+5))/(x+1)^(3)
((x-1)^(2)×(x-5))/(x+1)^(3)
((x-1)^(2)×(x+5))/(x-1)^(3)

The derivative of the function/ ((x-1)^(2)×(x+5))/(x+1)^(3)

Derivative of ((x-1)^(2)×(x+5))/(x+1)^(3)

The solution

You have entered [src]

       2        
(x - 1) *(x + 5)
----------------
           3    
    (x + 1)

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

((x - 1)^2*(x + 5))/(x + 1)^3

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \left(x - 1\right)^{2} \left(x + 5\right)$ and $g{\left(x \right)} = \left(x + 1\right)^{3}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \left(x - 1\right)^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = x - 1$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 1\right)$ :
    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: $1$
    The result of the chain rule is:
    
    $2 x - 2$
  $g{\left(x \right)} = x + 5$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x + 5$ term by term:
    1. The derivative of the constant $5$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result is: $\left(x - 1\right)^{2} + \left(x + 5\right) \left(2 x - 2\right)$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x + 1$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
  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: $1$
  The result of the chain rule is:
  
  $3 \left(x + 1\right)^{2}$
Now plug in to the quotient rule:

$\frac{- 3 \left(x - 1\right)^{2} \left(x + 1\right)^{2} \left(x + 5\right) + \left(x + 1\right)^{3} \left(\left(x - 1\right)^{2} + \left(x + 5\right) \left(2 x - 2\right)\right)}{\left(x + 1\right)^{6}}$
Now simplify:

$\frac{24 \left(x - 1\right)}{x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1}$

The answer is:

$\frac{24 \left(x - 1\right)}{x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2                                 2        
(x - 1)  + (-2 + 2*x)*(x + 5)   3*(x - 1) *(x + 5)
----------------------------- - ------------------
                  3                         4     
           (x + 1)                   (x + 1)

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

Simplify

The second derivative [src]

  /                                   2        \
  |    3*(-1 + x)*(3 + x)   2*(-1 + x) *(5 + x)|
6*|1 - ------------------ + -------------------|
  |                2                     3     |
  \         (1 + x)               (1 + x)      /
------------------------------------------------
                           2                    
                    (1 + x)

$$\frac{6 \left(\frac{2 \left(x - 1\right)^{2} \left(x + 5\right)}{\left(x + 1\right)^{3}} - \frac{3 \left(x - 1\right) \left(x + 3\right)}{\left(x + 1\right)^{2}} + 1\right)}{\left(x + 1\right)^{2}}$$

Simplify

The third derivative [src]

   /               2                             \
   |     5*(-1 + x) *(5 + x)   9*(-1 + x)*(3 + x)|
12*|-4 - ------------------- + ------------------|
   |                  3                    2     |
   \           (1 + x)              (1 + x)      /
--------------------------------------------------
                            3                     
                     (1 + x)

$$\frac{12 \left(- \frac{5 \left(x - 1\right)^{2} \left(x + 5\right)}{\left(x + 1\right)^{3}} + \frac{9 \left(x - 1\right) \left(x + 3\right)}{\left(x + 1\right)^{2}} - 4\right)}{\left(x + 1\right)^{3}}$$

Simplify

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Derivative of ((x-1)^(2)×(x+5))/(x+1)^(3)

The graph:

Piecewise:

The solution