Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of (x+1)/x
Derivative of sen2x
Derivative of ln(ln(x))
Derivative of log(x)*cos(x)
Identical expressions
(three (x- three))/(x- one)^(three)
(3(x minus 3)) divide by (x minus 1) to the power of (3)
(three (x minus three)) divide by (x minus one) to the power of (three)
(3(x-3))/(x-1)(3)
3x-3/x-13
3x-3/x-1^3
(3(x-3)) divide by (x-1)^(3)
Similar expressions
(3(x+3))/(x-1)^(3)
(3(x-3))/(x+1)^(3)

The derivative of the function/ (3(x-3))/(x-1)^(3)

Derivative of (3(x-3))/(x-1)^(3)

The solution

You have entered [src]

3*(x - 3)
---------
        3
 (x - 1)

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

(3*(x - 3))/(x - 1)^3

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 3 x - 9$ and $g{\left(x \right)} = \left(x - 1\right)^{3}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $3 x - 9$ term by term:
  1. The derivative of the constant $-9$ 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: $3$
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:

$\frac{3 \left(x - 1\right)^{3} - 3 \left(x - 1\right)^{2} \left(3 x - 9\right)}{\left(x - 1\right)^{6}}$
Now simplify:

$\frac{6 \left(4 - x\right)}{\left(x - 1\right)^{4}}$

The answer is:

$\frac{6 \left(4 - x\right)}{\left(x - 1\right)^{4}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3       9*(x - 3)
-------- - ---------
       3           4
(x - 1)     (x - 1)

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

Simplify

The second derivative [src]

   /     2*(-3 + x)\
18*|-1 + ----------|
   \       -1 + x  /
--------------------
             4      
     (-1 + x)

$$\frac{18 \left(\frac{2 \left(x - 3\right)}{x - 1} - 1\right)}{\left(x - 1\right)^{4}}$$

Simplify

The third derivative [src]

   /    5*(-3 + x)\
36*|3 - ----------|
   \      -1 + x  /
-------------------
             5     
     (-1 + x)

$$\frac{36 \left(- \frac{5 \left(x - 3\right)}{x - 1} + 3\right)}{\left(x - 1\right)^{5}}$$

Simplify