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Derivative of:
Derivative of 2x²
Derivative of f(x)=1
Derivative of e^x*x^2
Derivative of 2*sqrt(x^3)
Identical expressions
(two *x+ one)/(x- two)
(2 multiply by x plus 1) divide by (x minus 2)
(two multiply by x plus one) divide by (x minus two)
(2x+1)/(x-2)
2x+1/x-2
(2*x+1) divide by (x-2)
Similar expressions
(x^2+2*x+1)/(x*(-2)+4)
(2*x+1)/(x+2)
(2*x-1)/(x-2)

The derivative of the function/ (2*x+1)/(x-2)

Derivative of (2*x+1)/(x-2)

The solution

You have entered [src]

2*x + 1
-------
 x - 2

$$\frac{2 x + 1}{x - 2}$$

(2*x + 1)/(x - 2)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 x + 1$ and $g{\left(x \right)} = x - 2$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $2 x + 1$ term by term:
  1. The derivative of the constant $1$ 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 $x - 2$ 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 answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  2     2*x + 1 
----- - --------
x - 2          2
        (x - 2)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify