# The Derivative Of Quotient Of Functions

## The mathematics lesson. Derivatives topic. The quotient rule in derivatives. Derivatives of the functions that in f(x) / g(x) form. The derivative of the division of two functions.

If f(x) and g(x) are two functions and g(x) )≠ 0, the derivative of the f(x) / g(x) is

[ | f(x) |
| ||||

g(x) |

**Example:**

A function f(x) has given in the form at following.

f(x) = 1/x

Find the derivative of the function f(x).

**Solution:**

Let we partition the function f(x) to g(x) and h(x) functions.

Suppose that the function in the numerator is g(x) and the function in the denominator is h(x).

g(x) = 1

g’(x) = 0

h(x) = x

h’(x) =1

f(x) = | g(x) | |

h(x) |

f’(x) = | g’(x) . h(x) – h’(x) . g(x) | |

[g(x)]^{2} |

f’(x) = | 0 – 1 | |

x^{2} |

f’(x) = – | 1 | |

x^{2} |

**Example:**

A function f(x) has given in the form at following.

f(x) = | x – 1 | |

x + 1 |

Find the derivative of the function f(x).

**Solution:**

Suppose that the function in the numerator is g(x) , and the function in the denominator is h(x) .

f(x) = | g(x) | |

h(x) |

g(x) = x – 1

h(x) = x + 1

g’(x) = 1

h’(x) = 1.

The derivative of division of two function is given in the form below.

f’(x) = | g’(x) . h(x) – h’(x) . g(x) | |

[h(x)]^{2} |

f’(x) = | 1.(x + 1) – 1.(x – 1) | |

(x + 1)^{2} |

f’(x) = | 2 | |

(x + 1)^{2 } |

**Example:**

The functions f(x) and g(x) are defined in the cluster of the real numbers.

f(x) = x^{2} + 14x + 48

g(x) = x^{2} + 2x + 48

Calculate the [ | f(x) | ]’ |

g(x) |

**Solution:**

y = | f(x) | |

g(x) |

y = | x^{2} + 14x + 48 | |

x^{2} + 2x + 48 |

Let we exam whether f (x) and g (x) functions can be separated into its multipliers.

f(x) = (x + 8)(x + 6)

g(x) = (x – 6)(x + 8)

Both functions can be seperated into its multipliers. Let we simplfy its now.

f(x) |
| |||||

g(x) |

y = | (x + 6) | |

(x – 6) |

Now, let we calculate its derivates.

y' = | (x + 6)’ . (x – 6) – (x – 6)’ . (x + 6) | |

(x – 6)^{2} |

y' = | 1.(x – 6) – 1.(x + 6) | |

(x – 6)^{2} |

y' = | x – 6 – x – 6 | |

(x – 6)^{2} |

y’ = - | 12 | |

(x – 6)^{2} |

**Example:**

k(x) = | x^{5} + 2x^{2} | |

x^{2} + 3x |

Find the value of the k’(2).

**Solution:**

k(x) = | x^5 + 2x^2 | |

x^2 + 3x |

k(x) = | x(x^{4} + 2x) | |

x(x + 3) |

k(x) = | x^{4} + 2x | |

x + 3 |

k’(x) = | (x^{4} + 2x)’ . (x + 3) – (x + 3)’ . (x^{4} + 2x) | |

(x + 3)^{2} |

k’(x) = | (4x^{3} + 2) . (x + 3) – 1.(x^{4} + 2x) | |

(x + 3) |

k’(x) = | 4x^{4}+ 12x^{3} + 2x + 6 – x^{4} – 2x | |

(x + 3)^{2} |

k’(x) = | 3x^{4} + 12x^{3 } + 6 | |

(x + 3)^{2} |

k'(2) = | 3.2^{4} + 12.2^{3}+ 6 | |

5^{2} |

k'(2) = 6

**Example:**

The function f(x) has given in the form at following.

y = | 3x^{2} + 12x | |

6x+ 7 |

Calculate the | dy | (1) |

dx |

**Solution:**

Suppose that

y = | h(x) | |

k(x) |

The function hx is

h(x) = 3x^{2} + 12x

The function k(x) is

k(x) = 6x + 7

Derivative of the function h(x) is

h’(x) = 6x + 12

Derivative of the function k(x) is

k’(x) = 6

The rule of the division in the derivates.

y' = | h’(x) . k(x) – k’(x) . h(x) | |

[k(x)]^{2} |

y' = | (6x + 12) . (6x + 7) – 6(3x^{2} + 12x) | |

(6x + 7)^{2} |

y' = | 36x^{2} + 42x + 72x + 84 – 18x^{2} – 72x | |

(6x + 7)^{2} |

y' = | 18x^{2} + 42x + 84 | |

(6x + 7)^{2} |

dy |
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dx |

dy |
| |||||

dx |

RISE KNOWLEDGE

February 5 2019

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