The quotient rule

For fractions of two functions of x there is the quotient rule. The order in the numerator matters, otherwise the sign flips.

Rules

Three problems on this topic. Type like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x).

Task 1 of 3Quotient rule

Task

Differentiate:

f (x) = \frac{4 x - 3}{x - 1}

$f^{'} (x) =$

Write like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x), sin(x). Use brackets: e^(2x), not e^2x. Numbers also as 3/4 or e^2 - 1.

Worked example

Differentiate: $f (x) = \frac{4 x + 2}{x - 3}$

Worked solution

1.Quotient rule: $(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}$
2. $u = 4 x + 2, u^{'} = 4$ · $v = x - 3, v^{'} = 1$
3. $f^{'} (x) = \frac{4 ( x - 3 ) - ( 4 x + 2 ) \cdot 1}{( x - 3 ) ^{2}} = \frac{- 14}{( x - 3 ) ^{2}}$

Typical mistake: Computing $u v^{'} - u^{'} v$ in the numerator (wrong order flips the sign).

Common questions

Yes: u/v = u · v^(−1), then product and chain rule. The result is the same, it is just often more work to simplify.