The quotient rule expresses the derivative of f(x)/g(x), where f and g are differentiable and g(x)=0, as [g(x)]2g(x)f′(x)−f(x)g′(x). 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.
Detailed Explanation: Differentiate quotients using the quotient rule
To differentiate a quotient, use the quotient rule:
(g(x)f(x))′=[g(x)]2g(x)f′(x)−f(x)g′(x),g(x)=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=3x−2x2+1.
Step 1: Identify the numerator and denominator
Let
f(x)=x2+1andg(x)=3x−2.
Step 2: Find their derivatives
f′(x)=2xandg′(x)=3.
Step 3: Substitute into the quotient rule
y′=(3x−2)2(3x−2)(2x)−(x2+1)(3).
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(x2+1)=6x2−4x−3x2−3=3x2−4x−3.
Therefore,
y′=(3x−2)23x2−4x−3.
The original function, and therefore its derivative formula, is defined only when 3x−2=0, so x=32.
Learn by doing: Differentiate quotients using the quotient rule
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Practice:
Derivative Rules - Quotient Rule Positive Powers to Derivative