Product and Quotient Rule

Front

What is the Quotient Rule in calculus?

Back

The Quotient Rule states that if you have a function that is the quotient of two other functions, say \( f(x) = \frac{g(x)}{h(x)} \), then the derivative is given by \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \).

Front

What is the Product Rule in calculus?

Back

The Product Rule states that if you have a function that is the product of two other functions, say \( f(x) = g(x)h(x) \), then the derivative is given by \( f'(x) = g'(x)h(x) + g(x)h'(x) \).

Front

How do you apply the Quotient Rule?

Back

To apply the Quotient Rule, identify the numerator and denominator functions, find their derivatives, and then use the formula \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \).

Front

Find the derivative of \( y = \frac{x^2}{3x-1} \) using the Quotient Rule.

Back

The derivative is \( y' = \frac{3x^2 - 2x}{(3x-1)^2} \).

Front

Find the derivative of \( g(x) = \frac{3x-2}{x^2+2} \) using the Quotient Rule.

Back

The derivative is \( g'(x) = \frac{3(x^2+2) - (3x-2)(2x)}{(x^2+2)^2} = \frac{3x^2 + 6 - 6x^2 + 4x}{(x^2+2)^2} = \frac{-3x^2 + 4x + 6}{(x^2+2)^2} \).

Front

Find the derivative of \( y = \frac{4x^2 + 3}{2x^2 - 5} \) using the Quotient Rule.

Back

The derivative is \( y' = \frac{-52x}{(2x^2-5)^2} \).

Front

Use the Product Rule to find the derivative of \( f(x) = -x^3(3x^4 - 2) \).

Back

The derivative is \( f'(x) = -21x^6 + 6x^2 \).

Front

What is the first step in applying the Product Rule?

Back

Identify the two functions being multiplied and label them as \( g(x) \) and \( h(x) \).

Front

What is the significance of the Quotient Rule?

Back

The Quotient Rule is significant because it allows us to differentiate functions that are expressed as the ratio of two other functions, which is common in calculus.

Front

Provide an example of a function that requires the Product Rule to differentiate.

Back

An example is \( f(x) = (2x^3)(x^2 + 1) \).