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Derivative of (5x^2+2)×(x^2-4)^3
Identical expressions
y=((x- three)/(x+ two))^ three
y equally ((x minus 3) divide by (x plus 2)) cubed
y equally ((x minus three) divide by (x plus two)) to the power of three
y=((x-3)/(x+2))3
y=x-3/x+23
y=((x-3)/(x+2))³
y=((x-3)/(x+2)) to the power of 3
y=x-3/x+2^3
y=((x-3) divide by (x+2))^3
Similar expressions
y=((x+3)/(x+2))^3
y=((x-3)/(x-2))^3

The derivative of the function/ y=((x-3)/(x+2))^3

Derivative of y=((x-3)/(x+2))^3

The solution

You have entered [src]

       3
/x - 3\ 
|-----| 
\x + 2/

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

((x - 3)/(x + 2))^3

Detail solution

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

$\frac{15 \left(x - 3\right)^{2}}{\left(x + 2\right)^{2} \left(x + 2\right)^{2}}$
Now simplify:

$\frac{15 \left(x - 3\right)^{2}}{\left(x + 2\right)^{4}}$

The answer is:

$\frac{15 \left(x - 3\right)^{2}}{\left(x + 2\right)^{4}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       3                            
(x - 3)          /  3     3*(x - 3)\
--------*(x + 2)*|----- - ---------|
       3         |x + 2           2|
(x + 2)          \         (x + 2) /
------------------------------------
               x - 3

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                /                                    /     -3 + x\         \
                |                2                 3*|-1 + ------|*(-3 + x)|
  /     -3 + x\ |     17*(-3 + x)    13*(-3 + x)     \     2 + x /         |
3*|-1 + ------|*|-2 - ------------ + ----------- - ------------------------|
  \     2 + x / |              2        2 + x               2 + x          |
                \       (2 + x)                                            /
----------------------------------------------------------------------------
                                         3                                  
                                  (2 + x)

$$\frac{3 \left(\frac{x - 3}{x + 2} - 1\right) \left(- \frac{17 \left(x - 3\right)^{2}}{\left(x + 2\right)^{2}} - \frac{3 \left(x - 3\right) \left(\frac{x - 3}{x + 2} - 1\right)}{x + 2} + \frac{13 \left(x - 3\right)}{x + 2} - 2\right)}{\left(x + 2\right)^{3}}$$

Simplify

The graph

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Derivative of y=((x-3)/(x+2))^3

The graph:

Piecewise:

The solution