Derivative 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 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

Differentiate the function \( y = \frac{x^2 + 6}{2x - 7} \).

Back

Using the Quotient Rule, \( y' = \frac{(2x)(2x - 7) - (x^2 + 6)(2)}{(2x - 7)^2} = \frac{2x^2 - 14x - 12}{(2x - 7)^2} \).

Front

Find the derivative of \( f(x) = \frac{2x - 7}{x + 3} \).

Back

Using the Quotient Rule, \( f'(x) = \frac{(2)(x + 3) - (2x - 7)(1)}{(x + 3)^2} = \frac{13}{(x + 3)^2} \).

Front

What is the first step in applying the Quotient Rule?

Back

Identify the numerator and denominator functions, \( g(x) \) and \( h(x) \), respectively, in the quotient \( f(x) = \frac{g(x)}{h(x)} \).

Front

Differentiate \( y = \frac{2}{2x^4 - 5} \).

Back

Using the Quotient Rule, \( y' = \frac{0(2x^4 - 5) - 2(8x^3)}{(2x^4 - 5)^2} = \frac{-16x^3}{(2x^4 - 5)^2} \).

Front

What is the derivative of a constant?

Back

The derivative of a constant is zero. For example, if \( c \) is a constant, then \( \frac{d}{dx}(c) = 0 \).

Front

Differentiate \( y = \frac{x^2}{3x - 1} \).

Back

Using the Quotient Rule, \( y' = \frac{(2x)(3x - 1) - (x^2)(3)}{(3x - 1)^2} = \frac{3x^2 - 2x}{(3x - 1)^2} \).

Front

What is the significance of the denominator in the Quotient Rule?

Back

The denominator \( (h(x))^2 \) in the Quotient Rule ensures that the derivative is defined and accounts for the rate of change of the denominator function.

Front

State the formula for the derivative of a quotient.

Back

If \( f(x) = \frac{g(x)}{h(x)} \), then \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \).

Front

What is the derivative of \( y = \frac{1}{x} \)?

Back

Using the Quotient Rule, \( y' = \frac{0 \cdot x - 1 \cdot 1}{x^2} = -\frac{1}{x^2} \).