Differentiation Rules Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium

Use the quotient rule to differentiate

f(x) = \dfrac{x^2 + 1}{x - 3}

Solution

1
Identify numerator $f = x^2+1$ and denominator $g = x-3$ .
2
Compute derivatives: $f' = 2x$ , $g' = 1$ .
3
Apply the quotient rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$ .
4
Substitute: $\frac{2x(x-3) - (x^2+1)(1)}{(x-3)^2}$ .
5
Expand the numerator: $2x^2 - 6x - x^2 - 1 = x^2 - 6x - 1$ .
6
Final answer: $\frac{x^2 - 6x - 1}{(x-3)^2}$ .

Answer

f'(x) = \frac{x^2 - 6x - 1}{(x-3)^2}

The quotient rule formula is 'lo d-hi minus hi d-lo over lo squared.' The numerator requires careful expansion — subtract the entire second term. A common sign error is forgetting that