Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x^2*sin(2*x-3)
Derivative of (x+1)^2*(x-5)-6
Derivative of log(11*x)
Derivative of log(t-1)
Identical expressions
y=(e^(x- one))/(x- one)
y equally (e to the power of (x minus 1)) divide by (x minus 1)
y equally (e to the power of (x minus one)) divide by (x minus one)
y=(e(x-1))/(x-1)
y=ex-1/x-1
y=e^x-1/x-1
y=(e^(x-1)) divide by (x-1)
Similar expressions
y=(e^(x-1))/(x+1)
y=(e^(x+1))/(x-1)

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

Derivative of y=(e^(x-1))/(x-1)

The solution

You have entered [src]

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

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

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

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 $g{\left(x \right)} = x - 1$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=(e^(x-1))/(x-1)

The graph:

Piecewise:

The solution