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Identical expressions
f(x)=(x²- one)³/(4x³- five)²
f(x) equally (x² minus 1)³ divide by (4x³ minus 5)²
f(x) equally (x² minus one)³ divide by (4x³ minus five)²
fx=x²-1³/4x³-5²
f(x)=(x²-1)³ divide by (4x³-5)²
Similar expressions
f(x)=(x²+1)³/(4x³-5)²
f(x)=(x²-1)³/(4x³+5)²

The derivative of the function/ f(x)=(x²-1)³/(4x³-5)²

Derivative of f(x)=(x²-1)³/(4x³-5)²

The solution

You have entered [src]

         3 
 / 2    \  
 \x  - 1/  
-----------
          2
/   3    \ 
\4*x  - 5/

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

(x^2 - 1)^3/(4*x^3 - 5)^2

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \left(x^{2} - 1\right)^{3}$ and $g{\left(x \right)} = \left(4 x^{3} - 5\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x^{2} - 1$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} - 1\right)$ :
  1. Differentiate $x^{2} - 1$ term by term:
    1. The derivative of the constant $-1$ is zero.
    2. Apply the power rule: $x^{2}$ goes to $2 x$
    The result is: $2 x$
  The result of the chain rule is:
  
  $6 x \left(x^{2} - 1\right)^{2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 4 x^{3} - 5$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x^{3} - 5\right)$ :
  1. Differentiate $4 x^{3} - 5$ 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: $x^{3}$ goes to $3 x^{2}$
      So, the result is: $12 x^{2}$
    The result is: $12 x^{2}$
  The result of the chain rule is:
  
  $12 x^{2} \left(8 x^{3} - 10\right)$
Now plug in to the quotient rule:

$\frac{- 12 x^{2} \left(x^{2} - 1\right)^{3} \left(8 x^{3} - 10\right) + 6 x \left(x^{2} - 1\right)^{2} \left(4 x^{3} - 5\right)^{2}}{\left(4 x^{3} - 5\right)^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                3               2
      2 / 2    \        / 2    \ 
  24*x *\x  - 1/    6*x*\x  - 1/ 
- --------------- + -------------
              3                2 
    /   3    \       /   3    \  
    \4*x  - 5/       \4*x  - 5/

$$- \frac{24 x^{2} \left(x^{2} - 1\right)^{3}}{\left(4 x^{3} - 5\right)^{3}} + \frac{6 x \left(x^{2} - 1\right)^{2}}{\left(4 x^{3} - 5\right)^{2}}$$

Simplify

The second derivative [src]

            /                                           2 /           3  \\
            |                                  /      2\  |       18*x   ||
            |                              8*x*\-1 + x / *|-1 + ---------||
            |                3 /      2\                  |             3||
  /      2\ |        2   48*x *\-1 + x /                  \     -5 + 4*x /|
6*\-1 + x /*|-1 + 5*x  - --------------- + -------------------------------|
            |                       3                         3           |
            \               -5 + 4*x                  -5 + 4*x            /
---------------------------------------------------------------------------
                                           2                               
                                /        3\                                
                                \-5 + 4*x /

$$\frac{6 \left(x^{2} - 1\right) \left(- \frac{48 x^{3} \left(x^{2} - 1\right)}{4 x^{3} - 5} + 5 x^{2} + \frac{8 x \left(x^{2} - 1\right)^{2} \left(\frac{18 x^{3}}{4 x^{3} - 5} - 1\right)}{4 x^{3} - 5} - 1\right)}{\left(4 x^{3} - 5\right)^{2}}$$

Simplify

The third derivative [src]

   /                           3 /           3            6   \                                                                  \
   |                  /      2\  |      108*x        864*x    |                                                2 /           3  \|
   |                2*\-1 + x / *|1 - --------- + ------------|                                     2 /      2\  |       18*x   ||
   |                             |            3              2|                                 36*x *\-1 + x / *|-1 + ---------||
   |                             |    -5 + 4*x    /        3\ |       2 /      2\ /        2\                    |             3||
   |  /        2\                \                \-5 + 4*x / /   18*x *\-1 + x /*\-1 + 5*x /                    \     -5 + 4*x /|
24*|x*\-3 + 5*x / - ------------------------------------------- - --------------------------- + ---------------------------------|
   |                                         3                                     3                                3            |
   \                                 -5 + 4*x                              -5 + 4*x                         -5 + 4*x             /
----------------------------------------------------------------------------------------------------------------------------------
                                                                      2                                                           
                                                           /        3\                                                            
                                                           \-5 + 4*x /

$$\frac{24 \left(\frac{36 x^{2} \left(x^{2} - 1\right)^{2} \left(\frac{18 x^{3}}{4 x^{3} - 5} - 1\right)}{4 x^{3} - 5} - \frac{18 x^{2} \left(x^{2} - 1\right) \left(5 x^{2} - 1\right)}{4 x^{3} - 5} + x \left(5 x^{2} - 3\right) - \frac{2 \left(x^{2} - 1\right)^{3} \left(\frac{864 x^{6}}{\left(4 x^{3} - 5\right)^{2}} - \frac{108 x^{3}}{4 x^{3} - 5} + 1\right)}{4 x^{3} - 5}\right)}{\left(4 x^{3} - 5\right)^{2}}$$

Simplify

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Derivative of f(x)=(x²-1)³/(4x³-5)²

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The solution