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The derivative of the function/ (5x-3)/(x+3)

Derivative of (5x-3)/(x+3)

The solution

You have entered [src]

5*x - 3
-------
 x + 3

$$\frac{5 x - 3}{x + 3}$$

(5*x - 3)/(x + 3)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 5 x - 3$ and $g{\left(x \right)} = x + 3$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $5 x - 3$ 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: $5$
  The result is: $5$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x + 3$ term by term:
  1. The derivative of the constant $3$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

$\frac{18}{\left(x + 3\right)^{2}}$

The answer is:

$\frac{18}{\left(x + 3\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  5     5*x - 3 
----- - --------
x + 3          2
        (x + 3)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify