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Identical expressions
y=(x³+3x)²/(3x- five)
y equally (x³ plus 3x)² divide by (3x minus 5)
y equally (x³ plus 3x)² divide by (3x minus five)
y=x³+3x²/3x-5
y=(x³+3x)² divide by (3x-5)
Similar expressions
y=(x³-3x)²/(3x-5)
y=(x³+3x)²/(3x+5)

The derivative of the function/ y=(x³+3x)²/(3x-5)

Derivative of y=(x³+3x)²/(3x-5)

The solution

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          2
/ 3      \ 
\x  + 3*x/ 
-----------
  3*x - 5

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x^{3} + 3 x$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{3} + 3 x\right)$ :
  1. Differentiate $x^{3} + 3 x$ term by term:
    1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
    2. 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: $3$
    The result is: $3 x^{2} + 3$
  The result of the chain rule is:
  
  $\left(3 x^{2} + 3\right) \left(2 x^{3} + 6 x\right)$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $3 x - 5$ term by term:
  1. The derivative of the constant $-5$ is zero.
  2. 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: $3$
  The result is: $3$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /    /          2                \                                  2                         \
  |    |  /     2\       2 /     2\|                        2 /     2\         /     2\ /     2\|
  |  9*\3*\1 + x /  + 2*x *\3 + x //       /       2\   27*x *\3 + x /    54*x*\1 + x /*\3 + x /|
6*|- ------------------------------- + 4*x*\6 + 5*x / - --------------- + ----------------------|
  |              -5 + 3*x                                           3                    2      |
  \                                                       (-5 + 3*x)           (-5 + 3*x)       /
-------------------------------------------------------------------------------------------------
                                             -5 + 3*x

$$\frac{6 \left(- \frac{27 x^{2} \left(x^{2} + 3\right)^{2}}{\left(3 x - 5\right)^{3}} + 4 x \left(5 x^{2} + 6\right) + \frac{54 x \left(x^{2} + 1\right) \left(x^{2} + 3\right)}{\left(3 x - 5\right)^{2}} - \frac{9 \left(2 x^{2} \left(x^{2} + 3\right) + 3 \left(x^{2} + 1\right)^{2}\right)}{3 x - 5}\right)}{3 x - 5}$$

Simplify

The graph

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Derivative of y=(x³+3x)²/(3x-5)

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