Mastering the Quotient Rule for Finding Derivatives of Functions: A Comprehensive Guide

d/dx (f/g)

(gf’-fg’)/g^2

To find the derivative of the quotient f(x)/g(x) with respect to x, we can use the quotient rule:
“`
(d/dx) (f(x)/g(x)) = [g(x)*d/dx(f(x)) – f(x)*d/dx(g(x))] / [g(x)]^2
“`
Therefore:
“`
d/dx (f/g) = [g*d/dx(f) – f*d/dx(g)] / g^2
“`
Alternatively, we can rewrite f/g as f * (1/g) and use the product rule:
“`
d/dx (f/g) = d/dx(f * (1/g)) = d/dx(f) * (1/g) + f * d/dx(1/g)
“`
To find d/dx(1/g), we can use the chain rule:
“`
d/dx(1/g) = -1/g^2 * d/dx(g)
“`
Therefore, substituting, we get:
“`
d/dx (f/g) = d/dx(f) * (1/g) – f * d/dx(g) / g^2
“`
Both methods should give the same answer, and the choice of which one to use depends on personal preference or ease of calculation.

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