Introduction
A rational function is a function that can be expressed as the quotient of two polynomials. Polynomials are expressions that consist of variables and coefficients, which are multiplied or added together. A rational function can be written in the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) is not equal to zero.
To find the derivative of a rational function, we need to use the quotient rule, which is a formula used to find the derivative of a quotient of two functions. The quotient rule states that the derivative of f(x)/g(x) is equal to (g(x)*f'(x) - f(x)*g'(x))/g(x)^2, where f'(x) and g'(x) are the derivatives of f(x) and g(x), respectively.
Finding the Derivative of a Rational Function
Let's take the rational function f(x) = (2x^2 + 3x - 4)/(x^2 + 1) as an example. To find the derivative of this function, we need to use the quotient rule. First, we need to find the derivatives of the numerator and denominator separately.
The derivative of the numerator is f'(x) = 4x + 3, and the derivative of the denominator is g'(x) = 2x. Now we can plug these values into the quotient rule formula and simplify:
f'(x) = [(x^2 + 1)*(4x + 3) - (2x^2 + 3x - 4)*(2x)]/(x^2 + 1)^2
After simplifying, we get f'(x) = (5x^2 + 6x + 4)/(x^2 + 1)^2. This is the derivative of the rational function f(x) = (2x^2 + 3x - 4)/(x^2 + 1).
General Rule for Finding the Derivative of a Rational Function
The general rule for finding the derivative of a rational function is similar to the example above. Let f(x) = p(x)/q(x) be a rational function, where p(x) and q(x) are polynomials and q(x) is not equal to zero. The derivative of f(x) can be found using the quotient rule:
After simplifying, we get f'(x) = (q(x)*p'(x) - p(x)*q'(x))/q(x)^2. This is the derivative of the rational function f(x) = p(x)/q(x).
Examples of Finding the Derivative of a Rational Function
Let's take some more examples to illustrate the general rule for finding the derivative of a rational function.
Example 1
Find the derivative of f(x) = (3x^2 - 2x + 1)/(x^3 + 2x^2 - x - 2).
After simplifying, we get f'(x) = (10x^3 - 15x^2 - 8x + 2)/(x^3 + 2x^2 - x - 2)^2.
Example 2
Find the derivative of f(x) = (x^2 - 4)/(x^3 - 3x^2 + 2x).
After simplifying, we get f'(x) = (-x^4 + 10x^3 - 16x^2 + 8x + 16)/(x^3 - 3x^2 + 2x)^2.
Conclusion
The derivative of a rational function can be found using the quotient rule, which is a formula used to find the derivative of a quotient of two functions. To find the derivative of a rational function, we need to find the derivatives of the numerator and denominator separately, then plug them into the quotient rule formula and simplify. The general rule for finding the derivative of a rational function is similar to the example above. We can take some examples to illustrate the general rule for finding the derivative of a rational function.
