Quotient Rule

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, \( \frac{f}{g} \), then its derivative is given by \( \frac{f'g - fg'}{g^2} \).

Front

What is the derivative of a constant function?

Back

The derivative of a constant function is 0.

Front

How do you find the derivative of \( g(x) = \frac{3x-2}{x^2+2} \)?

Back

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

Front

What is the derivative of \( y = 2\pi \)?

Back

The derivative is 0, since 2π is a constant.

Front

What is the derivative of \( x^{-1} \)?

Back

The derivative is \( -x^{-2} \) or \( \frac{-1}{x^2} \).

Front

If \( f(x) = \frac{u}{v} \), what are u and v in the context of the Quotient Rule?

Back

In the context of the Quotient Rule, u is the numerator function and v is the denominator function.

Front

What is the formula for the derivative of a product of two functions?

Back

The Product Rule states that if \( f(x) = u(x)v(x) \), then \( f'(x) = u'v + uv' \).

Front

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

Back

The denominator \( g^2 \) in the Quotient Rule ensures that the derivative is defined wherever g is not zero.

Front

How do you apply the Quotient Rule to find the derivative of \( \frac{f(x)}{g(x)} \)?

Back

Differentiate the numerator and denominator separately, then apply the formula: \( \frac{f'g - fg'}{g^2} \).

Front

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

Back

The derivative is \( -\frac{1}{x^2} \).