Quantitative Aptitude · Differential and Integral Calculus

Differentiation Rules and Standard Formulas for CA Foundation

Updated 1 October 2026 · Fact-checked

Differentiation finds the rate at which a function changes, written dy/dx. You solve questions by spotting the function type, then applying the power, sum, product, quotient or chain rule with standard derivatives of xⁿ, eˣ and log x. Simplify, then substitute any given value of x.

Understand Differentiation Rules and Standard Formulas

The derivative of y = f(x) tells you how fast y changes when x changes by a tiny amount. Think of speed: distance changes with time, and speed is the rate of that change. In business, cost changes with output, and the derivative of cost is the extra cost of one more unit.

From first principles, the derivative is defined as a limit: f'(x) = lim (h→0) [f(x + h) − f(x)] ÷ h. You take a small step h, find the change in f, divide by h, then let h shrink to zero. Exam questions rarely ask you to do this in full, but you should know the definition and be able to use it for simple functions like x² or a constant times x.

All the rules are shortcuts built from that definition. The power rule handles xⁿ. The sum rule lets you differentiate term by term. The product rule and quotient rule handle two functions multiplied or divided. The chain rule handles a function inside another function, like (3x + 2)⁵ or e^(x²).

You must also memorise a few standard derivatives: eˣ stays eˣ, log x gives 1/x, and aˣ gives aˣ log a. Here log means the natural logarithm (base e), as used in calculus. Once these are automatic, most questions take under a minute.

Key formulas to remember

Derivative from first principles: f'(x) = lim (h→0) [f(x + h) − f(x)] ÷ h; Use only when the question says 'from first principles' or 'by definition'.
Derivative of a constant: d/dx (c) = 0; Applies to any number such as 5, e or log 2.
Power rule: d/dx (xⁿ) = n·xⁿ⁻¹; Works for any real n, including negative and fractional. So d/dx (1/x) = −1/x² and d/dx (√x) = 1/(2√x).
Constant multiple and sum rule: d/dx [c·f(x) ± g(x)] = c·f'(x) ± g'(x); Differentiate each term separately.
Product rule: d/dx (u·v) = u·(dv/dx) + v·(du/dx); Do not multiply the two derivatives.
Quotient rule: d/dx (u ÷ v) = [v·(du/dx) − u·(dv/dx)] ÷ v²; Valid where v ≠ 0. Numerator order matters: v·u' first, then minus u·v'.
Chain rule: dy/dx = (dy/du) × (du/dx), where y = f(u) and u = g(x); Differentiate the outer function, keep the inner as it is, then multiply by the derivative of the inner.
Exponential derivatives: d/dx (eˣ) = eˣ; d/dx (aˣ) = aˣ·log a (a > 0); d/dx (e^(f(x))) = e^(f(x))·f'(x) by the chain rule.
Log derivatives: d/dx (log x) = 1/x (x > 0); d/dx (log f(x)) = f'(x) ÷ f(x); Here log is the natural logarithm. For base a, d/dx (logₐ x) = 1 ÷ (x log a).

How to solve Differentiation Rules and Standard Formulas questions

Use this routine for any differentiation question. It stops you picking the wrong rule.

1Rewrite the function in simple powers. Convert roots and fractions, for example 1/x³ becomes x⁻³ and √x becomes x^(1/2).
2Look at the structure. Is it a sum, a product, a quotient, or a function inside a function?
3Pick the rule: sum for + or −, product for u·v, quotient for u/v, chain for a bracket raised to a power or e or log of an expression.
4Write u and v (or the outer and inner function) and their derivatives separately before combining.
5Apply the rule carefully and keep brackets around each derivative.
6Simplify. Combine like terms and factor if the options look factorised.
7If a value of x is given, substitute only after you have simplified.
8Match your answer with the options and check the sign and the power.

Quickest way: Spot the type, then use the shortcut

When to use it: Use this in the MCQ paper when you have about a minute per question.

Before using the quotient rule, check whether you can split the fraction. For example (x² + 1)/x = x + 1/x, which is easier.
For a bracket raised to a power, use the shortcut: n × (bracket)ⁿ⁻¹ × derivative of the bracket.
For e^(f(x)), write the same expression back and multiply by f'(x). For log f(x), write f'(x) over f(x).
If a value of x is asked, substitute early only in the final derivative, not in the original function.
Eliminate options: check sign, the power of x and whether a factor from the chain rule is missing. Often two options fail this test.
If a question needs a long product or quotient with several terms, mark it and come back. Wrong answers cost 0.25 marks.

Common mistakes in Differentiation Rules and Standard Formulas

Writing d/dx (u·v) = u'·v'
Students assume the derivative of a product is the product of derivatives, as it is for a sum.
Fix: Always write u·v' + v·u'. Test with x·x: the rule gives 2x, while u'·v' gives 1.
Reversing the numerator in the quotient rule
The order v·u' − u·v' is easy to flip under pressure.
Fix: Remember 'low d-high minus high d-low, over low squared'. The denominator's derivative term always comes second.
Forgetting the inner derivative in the chain rule
Students differentiate only the outer function, for example d/dx (3x + 2)⁵ = 5(3x + 2)⁴.
Fix: Multiply by the derivative of the inside. The correct result is 5(3x + 2)⁴ × 3 = 15(3x + 2)⁴.
Differentiating eˣ or aˣ with the power rule
Students see a power and apply n·xⁿ⁻¹, giving x·eˣ⁻¹.
Fix: Check where the variable is. If x is in the exponent, use the exponential formula, not the power rule.
Writing the derivative of log x as log(1/x) or x
Mixing up differentiation with log laws.
Fix: Memorise d/dx (log x) = 1/x. For log f(x), write f'(x)/f(x).
Treating a constant like a variable, such as d/dx (e²) = 2e
The symbol e looks like a variable.
Fix: e², π, log 3 and similar are constants. Their derivative is 0.

Worked examples

Example 1

If y = x²·eˣ, then dy/dx equals: (A) x²eˣ (B) 2x·eˣ (C) eˣ(x² + 2x) (D) eˣ(x² + 2)

Show the solution

Use the product rule with u = x² and v = eˣ.
du/dx = 2x and dv/dx = eˣ.
dy/dx = u·v' + v·u' = x²·eˣ + eˣ·2x.
Factor out eˣ: dy/dx = eˣ(x² + 2x).

Answer: (C) eˣ(x² + 2x)

Example 2

If y = (x² + 1) ÷ (x − 1), then dy/dx at x = 2 equals: (A) −1 (B) 1 (C) 3 (D) −3

Show the solution

Use the quotient rule with u = x² + 1 and v = x − 1.
u' = 2x and v' = 1.
dy/dx = [v·u' − u·v'] ÷ v² = [(x − 1)(2x) − (x² + 1)(1)] ÷ (x − 1)².
Numerator = 2x² − 2x − x² − 1 = x² − 2x − 1.
At x = 2: numerator = 4 − 4 − 1 = −1, denominator = (1)² = 1.
So dy/dx = −1.

Answer: (A) −1

Example 3

If y = log(3x² + 1), then dy/dx at x = 1 equals: (A) 3/2 (B) 2 (C) 3 (D) 1/4

Show the solution

Use d/dx (log f(x)) = f'(x) ÷ f(x), with f(x) = 3x² + 1 and f'(x) = 6x.
dy/dx = 6x ÷ (3x² + 1).
At x = 1: 6 ÷ (3 + 1) = 6/4 = 3/2.

Answer: (A) 3/2

Exam tips

Expect direct one-step questions: find dy/dx for a polynomial, a product, a quotient or e/log of an expression. Practise speed on these.
Always simplify before substituting a value of x, and re-check the arithmetic, since options often contain sign traps.
Learn the chain rule forms e^(f(x)) → e^(f(x))·f'(x) and log f(x) → f'(x)/f(x). They appear again in maxima, minima and marginal analysis.
If the question has a fraction with a single-term denominator, split it into separate powers of x instead of using the quotient rule.
Revise limits as well, because first-principles questions use the limit definition.

Practice questions from Differential and Integral Calculus

Differentiation Rules and Standard Formulas: frequently asked questions

What are the most important differentiation formulas for CA Foundation?

Learn the power rule, the sum rule, the product and quotient rules, and the chain rule. Add the derivatives of eˣ, aˣ and log x. These cover almost every question in this chapter.

When do I use the quotient rule instead of the product rule?

Use the quotient rule when one expression is divided by another with x in both. You can also write u/v as u × v⁻¹ and use the product rule, but the quotient rule is usually quicker. If the denominator is a single term like x², split the fraction first.

What is the derivative of log x in CA Foundation maths?

It is 1/x, for x > 0. Here log means the natural logarithm. For log of an expression f(x), the derivative is f'(x) divided by f(x).

Do I need to learn differentiation from first principles?

Know the definition and how to apply it to simple functions such as x² or xⁿ. Most MCQs use the rules, but the definition helps you understand why the rules work.