Mister Exam

Lang:

The derivative of the function/ (4x-3)/(x-1)

Derivative of (4x-3)/(x-1)

The solution

You have entered [src]

4*x - 3
-------
 x - 1

$$\frac{4 x - 3}{x - 1}$$

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 4 x - 3$ and $g{\left(x \right)} = x - 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $4 x - 3$ term by term:
  1. The derivative of the constant $-3$ 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: $4$
  The result is: $4$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  4     4*x - 3 
----- - --------
x - 1          2
        (x - 1)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /    -3 + 4*x\
6*|4 - --------|
  \     -1 + x /
----------------
           3    
   (-1 + x)

$$\frac{6 \left(4 - \frac{4 x - 3}{x - 1}\right)}{\left(x - 1\right)^{3}}$$

Simplify