Explanation
Definition
The quotient rule is a differentiation rule used to differentiate fractions. It is applied whenever a function is presented as the quotient of two functions. The quotient rule formula is:
\left(\frac{u(x)}{v(x)}\right)' = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{[v(x)]^2}
Procedure
Scheme
To differentiate using the quotient rule, first identify the functions u(x) (numerator) and v(x) (denominator). Then determine their derivatives u'(x) and v'(x). Next, substitute these into the quotient rule formula.
f'(x) = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{[v(x)]^2}
Examples
Differentiate the function f(x) = \frac{x}{x+1}.
1. u(x) = x, v(x) = x+1
2. u'(x) = 1, v'(x) = 1
3. Substitute into the quotient rule formula:
f'(x) = \frac{1 \cdot (x+1) - x \cdot 1}{(x+1)^2} = \frac{1}{(x+1)^2}
Differentiate the function f(x) = \frac{2x^2}{3x-4}.
1. u(x) = 2x^2, v(x) = 3x-4
2. u'(x) = 4x, v'(x) = 3
3. Substitute into the quotient rule formula:
f'(x) = \frac{4x(3x-4) - 2x^2 \cdot 3}{(3x-4)^2} = \frac{12x^2 - 16x - 6x^2}{(3x-4)^2} = \frac{6x^2 - 16x}{(3x-4)^2}
Note
Summary
- ★The quotient rule is used to differentiate fractions.
- ★Remember the quotient rule formula: derivative of numerator times denominator minus numerator times derivative of denominator, divided by the denominator squared.
Exercises
1 / 3
Differentiate the function
f(x) = \frac{3x}{x-2}
2 / 3
Differentiate the function
f(x) = \frac{x^2+1}{2x+3}
3 / 3
Differentiate the function
f(x) = \frac{5x^3 - 2x}{x^2 + 4}
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