Quotient Rule

Front

What is the Quotient Rule in calculus?

Back

The Quotient Rule is a method for finding the derivative of a function that is the quotient of two other functions. It states that if you have a function f(x) = g(x)/h(x), then the derivative f'(x) is given by: f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2.

Front

When should you use the Quotient Rule?

Back

You should use the Quotient Rule when you need to differentiate a function that is expressed as the ratio of two differentiable functions.

Front

State the formula for the Quotient Rule.

Back

If f(x) = g(x)/h(x), then f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2.

Front

Differentiate the function f(x) = (2x^3 + 3)/(x^2 - 1).

Back

Using the Quotient Rule: f'(x) = [(6x^2)(x^2 - 1) - (2x^3 + 3)(2x)] / (x^2 - 1)^2.

Front

What are the components of the Quotient Rule formula?

Back

The components are: g(x) = the numerator function, h(x) = the denominator function, g'(x) = the derivative of the numerator, and h'(x) = the derivative of the denominator.

Front

Explain the significance of the denominator in the Quotient Rule.

Back

The denominator h(x) in the Quotient Rule must not be zero, as this would make the function undefined.

Front

Provide an example of a function where the Quotient Rule is applicable.

Back

An example is f(x) = (x^2 + 1)/(x - 3), where the numerator and denominator are both polynomials.

Front

What is the first step in applying the Quotient Rule?

Back

Identify the numerator g(x) and the denominator h(x) of the function you want to differentiate.

Front

How does the Quotient Rule relate to the Product Rule?

Back

The Quotient Rule can be derived from the Product Rule by rewriting the quotient as a product of the numerator and the reciprocal of the denominator.

Front

Differentiate the function f(x) = (x^2)/(x^3 + 1).

Back

Using the Quotient Rule: f'(x) = [(2x)(x^3 + 1) - (x^2)(3x^2)] / (x^3 + 1)^2.