Rational Functions Calculus

In calculus, we use both the differentiation and integration methods. In rational functions calculus both the integration and differentiation process plays an important role. Rational function consists of both the numerator and denominator value. Rational function is denoted as,

f(x) = (g (x)) / (h (x))

Here, g (x) = numerator value, h (x) = denominator value.

The denominator value h (x) is not equal to zero. It consists of all real numbers.

Example problems for rational function calculus

Rational function calculus problem 1:

Differentiate the given function f (x) = (1 / (2x))

Solution:

Given function is (1 / (2x))

Rearrange the given equation, we get

f (x) = - 2x -1

Differentiate with trespect to x, we get

f' (x) = (- 2 * - 1) (x -2)

= 2 / x^2

Answer:

The final answer is 2 / x^2

Rational function calculus problem 2:

Differentiate the given function f (x) = (1 / (4x^2 + 4))

Solution:

Given function is (1 / (4x^2 + 4))

Rearrange the given equation, we get

f (x) = (- 4x -2 - 4)

Differentiate with trespect to x, we get

f' (x) = (- 4 * - 2) (x -3) + 0

= 8 / x^3

Answer:

The final answer is 8 / x^3

Rational function calculus problem 3:

Integrate the given rational function ∫ (dx / x)

Solution:

Given integration function (dx / x)

Integate the given function with respect to x, we get

∫ (dx / x) = ∫ (1 / x) dx

= log x + c

Answer:

The final answer is log x+ c

Rational function calculus problem 4:

Integrate the given rational function ∫ (dx / (x - a))

Solution:

Given function is dx / (x - a)

Integrate the given function with respect to x, we get

∫ (dx / (x - a)) = ∫ (1 / (x - a)) dx

= log (x - a) + c

Answer:

The final answer  is log (x - a) + c

Practice problems for rational functions calculus

Rational function calculus problem 1:

Integrate the given rational function ∫ (dx / (x^2 - a^2))

Answer:

The final answer is (- tan h^-1 (x / a)) / (a)

Rational function calculus problem 2:

Integrate the given rational function ∫ (dx / (x + a))

Answer:

The final answer is log (a + x) + c



Rational function calculus problem 3:

Differentiate the given function f (x) = (1 / (x^4))

Answer:

The final answer is (- 4 / x^5)

Rational function calculus problem 4:

Differentiate the given function f (x) = (1 / (5x^7))

Answer:

The final answer is (- 7 / (5x^8))