Quotient Rule Derivatives

Front

What is the Quotient Rule for derivatives?

Back

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

Front

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

Back

\( g'(x) = \frac{-3x^2 + 4x + 6}{(x^2 + 2)^2} \) using the Quotient Rule.

Front

If \( y = \frac{3}{4+x^2} \), what is \( \frac{dy}{dx} \)?

Back

\( \frac{dy}{dx} = -\frac{6x}{(4+x^2)^2} \) using the Quotient Rule.

Front

Given \( y = \frac{x+2}{x-3} \), find \( y' \).

Back

\( y' = -\frac{5}{(x-3)^2} \) using the Quotient Rule.

Front

What is the first step in applying the Quotient Rule?

Back

Identify the numerator \( u(x) \) and the denominator \( v(x) \) of the function \( f(x) = \frac{u(x)}{v(x)} \).

Front

What does \( u' \) and \( v' \) represent in the Quotient Rule?

Back

\( u' \) is the derivative of the numerator \( u(x) \) and \( v' \) is the derivative of the denominator \( v(x) \).

Front

How do you find the derivative of a constant function?

Back

The derivative of a constant function is zero, i.e., if \( f(x) = c \), then \( f'(x) = 0 \).

Front

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

Back

The denominator \( v(x) \) must not be zero, as division by zero is undefined.

Front

What is the general form of the Quotient Rule?

Back

If \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) = \frac{u'v - uv'}{v^2} \).

Front

When applying the Quotient Rule, what should you do after finding \( u' \) and \( v' \)?

Back

Substitute \( u, u', v, \) and \( v' \) into the Quotient Rule formula to find \( f'(x) \).