Example 1: Chain rule with a logarithm
Find dy/dx if y = ln(3x + 1).
- Let
u = 3x + 1, soy = ln u dy/du = 1/u,du/dx = 3dy/dx = (1/u) × 3
Answer: dy/dx = 3 / (3x + 1)
Differentiation Rules (DSE M1): Product, Quotient and Chain Rule
M1 differentiation uses four rules (addition, product, quotient, chain) and six formulas: (xⁿ)′ = nxⁿ⁻¹, (eˣ)′ = eˣ, (ln x)′ = 1/x, (aˣ)′ = aˣ ln a, (loga x)′ = 1/(x ln a) and (C)′ = 0. Trigonometric functions are not part of M1.
Differentiation Formulas (DSE M1)
Power
(xⁿ)′ = nxⁿ⁻¹Exponential
(eˣ)′ = eˣ, (aˣ)′ = aˣ ln aLogarithm
(ln x)′ = 1/x, (loga x)′ = 1/(x ln a)Product rule
(uv)′ = u′v + uv′Quotient rule
(u/v)′ = (u′v − uv′) / v²Chain rule
dy/dx = (dy/du)(du/dx)Worked Examples — Differentiation of a Function
Example 2: Product rule with an exponential
Differentiate y = (x² + 1)e³ˣ.
u = x² + 1,u′ = 2x;v = e³ˣ,v′ = 3e³ˣdy/dx = 2x e³ˣ + (x² + 1)(3e³ˣ)= e³ˣ(3x² + 2x + 3)
Answer: dy/dx = e³ˣ(3x² + 2x + 3)
💡 Exam Traps & Secrets
⚠️ The inside derivative
(e³ˣ)′ = 3e³ˣ and (ln(3x + 1))′ = 3/(3x + 1). Leaving out the factor from the inside function is the most frequent M1 differentiation error.⚠️ aˣ is not a power function
(2ˣ)′ = 2ˣ ln 2, not x · 2ˣ⁻¹. The power rule applies only when the variable is the base.✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Product and chain rules)Differentiate y = x³ e^(−2x).
(Chain rule)Differentiate y = ln(x² + 4).
(Quotient rule)Differentiate y = (2x + 1)/(x² + 3).
Frequently Asked Questions
No. M1 differentiates algebraic, exponential and logarithmic functions only. Trigonometric functions appear in M2 calculus.
No. The curriculum notes list implicit differentiation, logarithmic differentiation, parametric equations and inverse functions as not required in M1.
Whenever one function is inside another, such as e to the power 3x or ln(3x + 1): differentiate the outside, then multiply by the derivative of the inside.