Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 1/tg(x)
Derivative of (1-x^2)^2
Derivative of y=x^3-8x^2+10x-4
Derivative of x/(x^2+4)
Identical expressions
(e^x- one)/(x- two)
(e to the power of x minus 1) divide by (x minus 2)
(e to the power of x minus one) divide by (x minus two)
(ex-1)/(x-2)
ex-1/x-2
e^x-1/x-2
(e^x-1) divide by (x-2)
Similar expressions
(e^x-1)/(x+2)
(e^x+1)/(x-2)

The derivative of the function/ (e^x-1)/(x-2)

Derivative of (e^x-1)/(x-2)

The solution

You have entered [src]

 x    
E  - 1
------
x - 2

$$\frac{e^{x} - 1}{x - 2}$$

(E^x - 1)/(x - 2)

Detail solution

Apply the quotient rule, which is:

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   x      x     
  e      E  - 1 
----- - --------
x - 2          2
        (x - 2)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of (e^x-1)/(x-2)

The graph:

Piecewise:

The solution