Advanced Differentiation Rules
Beyond the basic power and sum rules, calculus provides powerful rules for differentiating more complex functions.
The Product Rule
Used to find the derivative of a product of two functions, f(x) and g(x).
Formula: If h(x) = f(x)g(x), then h'(x) = f(x)g'(x) + g(x)f'(x)
In words: 'The first times the derivative of the second, plus the second times the derivative of the first.'
Example: Find the derivative of h(x) = x²sin(x).
f(x) = x², f'(x) = 2x
g(x) = sin(x), g'(x) = cos(x)
h'(x) = (x²)(cos(x)) + (sin(x))(2x)
The Quotient Rule
Used to find the derivative of a quotient of two functions, f(x) and g(x).
Formula: If h(x) = f(x)/g(x), then h'(x) = [g(x)f'(x) - f(x)g'(x)] / [g(x)]²
Mnemonic: 'Low D-high minus High D-low, square the bottom and away we go!' (where D means 'derivative of').
Example: Find the derivative of h(x) = x³ / (x+1).
f(x) = x³, f'(x) = 3x²
g(x) = x+1, g'(x) = 1
h'(x) = [(x+1)(3x²) - (x³)(1)] / (x+1)² = (3x³ + 3x² - x³) / (x+1)² = (2x³ + 3x²) / (x+1)²
The Chain Rule
Used to find the derivative of a composite function (a function inside another function), like f(g(x)).
Formula: If h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x)
In words: 'The derivative of the outside function (leaving the inside alone), times the derivative of the inside function.'
Example: Find the derivative of h(x) = (x² + 3)⁴.
The 'outside' function is (something)⁴. Its derivative is 4(something)³.
The 'inside' function is x² + 3. Its derivative is 2x.
h'(x) = 4(x² + 3)³ * (2x) = 8x(x² + 3)³