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Derivative of:
Derivative of 1/3*x^3
Derivative of x^2/(x^2+1)
Derivative of x*sqrt(3-x)
Derivative of (x^2+25)/x
Identical expressions
(two *x- nine)/(x- five)
(2 multiply by x minus 9) divide by (x minus 5)
(two multiply by x minus nine) divide by (x minus five)
(2x-9)/(x-5)
2x-9/x-5
(2*x-9) divide by (x-5)
Similar expressions
(2*x+9)/(x-5)
(2*x-9)/(x+5)

The derivative of the function/ (2*x-9)/(x-5)

Derivative of (2*x-9)/(x-5)

The solution

You have entered [src]

2*x - 9
-------
 x - 5

$$\frac{2 x - 9}{x - 5}$$

(2*x - 9)/(x - 5)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 x - 9$ and $g{\left(x \right)} = x - 5$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $2 x - 9$ 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: $2$
  The result is: $2$
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: $1$
Now plug in to the quotient rule:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  2     2*x - 9 
----- - --------
x - 5          2
        (x - 5)

$$\frac{2}{x - 5} - \frac{2 x - 9}{\left(x - 5\right)^{2}}$$

Simplify

The second derivative [src]

  /     -9 + 2*x\
2*|-2 + --------|
  \      -5 + x /
-----------------
            2    
    (-5 + x)

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

Simplify

The third derivative [src]

  /    -9 + 2*x\
6*|2 - --------|
  \     -5 + x /
----------------
           3    
   (-5 + x)

$$\frac{6 \left(2 - \frac{2 x - 9}{x - 5}\right)}{\left(x - 5\right)^{3}}$$

Simplify

The graph

Derivative of (2*x-9)/(x-5)