The Quotient Rule
When you have a fraction of functions, use the Quotient Rule. It's a bit messier than the product rule, but follows a similar pattern.
Quotient Rule
Quotient Rule
d/dx[f(x)/g(x)] = [f'(x)*g(x) - f(x)*g'(x)] / [g(x)]^2
Basic Quotient Rule
Medium
Find d/dx[x^2 / (x+1)]
1
Identify f and g
f(x) = x^2 (high), g(x) = x+1 (low)
2
Find derivatives
f'(x) = 2x, g'(x) = 1
3
Apply quotient rule
= [2x(x+1) - x^2(1)] / (x+1)^2
4
Simplify
= [2x^2 + 2x - x^2] / (x+1)^2 = (x^2 + 2x) / (x+1)^2
Answer:
(x^2 + 2x) / (x+1)^2
Practice
1. d/dx[sin(x)/x] = ?
Key Takeaways
Quotient Rule: (f/g)' = (f'g - fg')/g^2
"Low dee-high minus high dee-low, over low squared"
Often can avoid by rewriting as product with negative exponent