Chain, product and quotient rules · Year 13

In Year 13 you learn: the chain rule.

Most functions are built from simpler ones: multiplied together, divided, or one inside another. The chain, product and quotient rules let you differentiate them, using the standard derivatives of powers, eˣ, ln x and the trig functions.

Key facts

• Chain rule: dy/dx = dy/du × du/dx. For example, d/dx (3x + 1)^5 = 15(3x + 1)^4.
• d/dx e^(kx) = ke^(kx) and d/dx ln x = 1/x. So d/dx ln(f(x)) = (f′(x))/(f(x)).
• d/dx sin(kx) = k cos(kx), d/dx cos(kx) = −k sin(kx), d/dx tan x = sec^2 x (x in radians).
• Implicit: differentiate every term with respect to x, using d/dx(y^2) = 2y dy/dx. Parametric: dy/dx = (dy/dt)/(dx/dt).

Worked examples