Product, quotient and chain rules

Expanding (3x + 1)⁴ takes five terms of algebra. The chain rule gives its derivative, 12(3x + 1)³, in one line.

The product rule

The last unit warned that the derivative of a product is not the product of the derivatives. The correct rule is d/dx (f·g) = f′·g + f·g′. Think of a rectangle whose sides f and g are both growing. Its area f·g grows in two strips: one along side g of width Δf, and one along side f of width Δg, plus a tiny corner that vanishes in the limit. For f(x) = x² and g(x) = x³ + 1, the rule gives 2x(x³ + 1) + x²(3x²) = 5x⁴ + 2x. Expanding first, x⁵ + x², gives the same answer.

The quotient rule

For a quotient, d/dx (f/g) = (f′·g − f·g′)/g², wherever g is not zero. The order in the numerator matters because of the minus sign; a common memory aid is low d-high minus high d-low, over low squared. For x/(x + 1): f = x, f′ = 1, g = x + 1, g′ = 1, so the derivative is ((x + 1) − x)/(x + 1)² = 1/(x + 1)². At x = 1 that is 1/4. The power rule for negative exponents, which you used on 1/x, is a special case.

Sine and cosine

OpenStax works out the derivative of sine from the limit definition, using two limits from the limits course: (sin h)/h → 1 and (cos h − 1)/h → 0 as h → 0, with h in radians. The result is d/dx sin x = cos x, and similarly d/dx cos x = −sin x. You can see the first one in the lab below: the slope of sin x is 1 at x = 0, 0 at the top x = π/2, and −1 at x = π, exactly the values of cos x. These formulas hold only in radians.

The chain rule

Many functions are built by putting one function inside another: (3x + 1)⁴, sin(2x), √(x² + 9). The chain rule handles all of them: if y = f(g(x)), then y′ = f′(g(x))·g′(x). Differentiate the outside function, leave the inside alone, then multiply by the derivative of the inside. For (3x + 1)⁴, the outside is u⁴ with derivative 4u³ and the inside 3x + 1 has derivative 3, so the derivative is 4(3x + 1)³·3 = 12(3x + 1)³. In Leibniz notation, dy/dx = (dy/du)(du/dx), which looks like fractions cancelling and is a reliable way to remember it.

The inside derivative is the factor people forget. The derivative of sin(2x) is cos(2x)·2, so at x = 0 the slope is 2, twice the slope of sin x there, because the graph is squeezed horizontally by a factor of 2. For √(x² + 9) = (x² + 9)^1/2, the derivative is (1/2)(x² + 9)^−1/2·2x = x/√(x² + 9); at x = 4 that is 4/5.