Skill: Differentiate quotients using the quotient rule

Explanation and Free Practice Resources

The quotient rule expresses the derivative of $f(x)/g(x)$ , where $f$ and $g$ are differentiable and $g(x)\ne0$ , as $\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}$ . It captures how simultaneous changes in the numerator and denominator affect a quotient, including the importance of the subtraction order and the squared denominator; applications are limited to standard algebraic and transcendental functions rather than more advanced generalized derivative theories.

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Detailed Explanation: Differentiate quotients using the quotient rule

To differentiate a quotient, use the quotient rule:

\left(\frac{f(x)}{g(x)}\right)' = \frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}, \qquad g(x)\ne 0.

A helpful way to remember the order is:

bottom times derivative of top minus top times derivative of bottom, all over bottom squared.

Example

Find the derivative of

y=\frac{x^2+1}{3x-2}.

Step 1: Identify the numerator and denominator

Let

f(x)=x^2+1 \qquad\text{and}\qquad g(x)=3x-2.

Step 2: Find their derivatives

f'(x)=2x \qquad\text{and}\qquad g'(x)=3.

Step 3: Substitute into the quotient rule

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

Notice that the subtraction is important: it is

g(x)f'(x)-f(x)g'(x),

not the other way around. Also, the entire denominator is squared.

Step 4: Simplify the numerator

(3x-2)(2x)-3(x^2+1) = 6x^2-4x-3x^2-3 = 3x^2-4x-3.

Therefore,

\boxed{y'=\frac{3x^2-4x-3}{(3x-2)^2}}.

The original function, and therefore its derivative formula, is defined only when $3x-2\ne0$ , so $x\ne \frac23$ .

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