Quantitative Aptitude · Differential and Integral Calculus

Higher Order Derivatives for CA Foundation

Updated 1 October 2026 · Fact-checked

A higher order derivative is the derivative of a derivative. The second derivative d²y/dx² comes from differentiating f′(x) once more. To solve a question, differentiate step by step, simplify after each step, and then substitute the given value of x only at the end.

Understand Higher Order Derivatives

The first derivative f′(x) tells you the rate at which y changes as x changes. It is itself a function of x. So you can differentiate it again.

Differentiating f′(x) gives the second derivative, written f″(x) or d²y/dx². Differentiating again gives the third derivative f‴(x) or d³y/dx³. Continue the same way for the nth derivative.

There is no new rule here. You use the same rules you already know: power rule, product rule, quotient rule, chain rule, and the standard formulas for eˣ, log x and so on. The only skill is to do it repeatedly without errors.

Why does it matter? The sign of f″(x) tells you about the shape of the curve. It is used in the second derivative test for maxima and minima. If f′(x) = 0 and f″(x) < 0, you get a maximum. If f′(x) = 0 and f″(x) > 0, you get a minimum. It also shows how a marginal quantity such as marginal cost is itself changing.

In the exam, questions are usually direct: find d²y/dx² of a given function, find its value at a point, or show a relation between y and its derivatives.

Key formulas to remember

Second derivative: d²y/dx² = d/dx (dy/dx) = f″(x); Differentiate the first derivative once more with respect to x.
Third derivative: d³y/dx³ = d/dx (d²y/dx²) = f‴(x); Keep going the same way for higher orders.
Power rule repeated: If y = xⁿ, then y′ = n xⁿ⁻¹ and y″ = n(n − 1) xⁿ⁻² (the y″ formula holds for any real n, and for a positive integer n it is meaningful for n ≥ 2). In general, the kth derivative is n(n − 1)(n − 2)...(n − k + 1) xⁿ⁻ᵏ.; For a positive integer n, the nth derivative of xⁿ is n! (a constant) and the (n + 1)th derivative is 0. For n = 1, y″ = 0, which the formula also gives.
Exponential: If y = eᵃˣ, then dⁿy/dxⁿ = aⁿ eᵃˣ; Each differentiation multiplies by a.
Logarithm: If y = log x, then y′ = 1/x, y″ = −1/x², y‴ = 2/x³; Valid for x > 0. Write 1/x as x⁻¹ and use the power rule.
Second derivative test: f′(x) = 0 and f″(x) < 0 → maximum; f′(x) = 0 and f″(x) > 0 → minimum; If f″(x) = 0 the test is inconclusive.

How to solve Higher Order Derivatives questions

Use this method for any question on second or higher order derivatives.

1Rewrite the function so every term is in a form you can differentiate, for example 1/x² as x⁻² and √x as x^(1/2).
2Find the first derivative using the correct rule (power, product, quotient or chain).
3Simplify the first derivative fully before moving on.
4Differentiate the simplified result to get the second derivative. Repeat if a higher order is asked.
5If a value of x is given, substitute it only after you have the final derivative of the required order.
6Check the sign and the arithmetic once, then match with the options.

Quickest way: Pattern and elimination approach

When to use it: Use this in the MCQ paper when the function is a polynomial, an exponential or a simple log, where a pattern exists.

For a polynomial, differentiate term by term. Constants vanish at the first step, so ignore them.
For a polynomial of degree n, the nth derivative is a constant and the (n + 1)th is zero. Use this to eliminate options at once.
For eᵃˣ, write the answer directly as aⁿ eᵃˣ.
For log x, remember the sequence 1/x, −1/x², 2/x³. The sign alternates.
If the options differ in sign, check the sign of the second derivative first and eliminate.
If a question needs a long product rule and time is short, skip it and return later. A wrong answer costs 0.25 marks.

Common mistakes in Higher Order Derivatives

Substituting the value of x after the first derivative and then differentiating again.
Students want to simplify early.
Fix: Differentiating a number gives zero, which is wrong. Substitute only after the full derivative is found.
Writing d²y/dx² as (dy/dx)².
The notation looks like a square.
Fix: d²y/dx² means differentiate twice. (dy/dx)² means the first derivative squared. They are different.
Forgetting the chain factor, for example the derivative of e²ˣ taken as e²ˣ.
Students memorise the formula for eˣ only.
Fix: Multiply by the derivative of the power each time. For e²ˣ the second derivative is 4e²ˣ.
Sign error in the log derivative, giving y″ = 1/x² for y = log x.
The negative power is dropped.
Fix: Write x⁻¹, differentiate to −x⁻², and keep the minus sign.
Not simplifying the first derivative before the second differentiation.
Students rush into the quotient rule on a messy expression.
Fix: Simplify or rewrite first. Often a quotient becomes a sum of powers.
Concluding a maximum or minimum when f″(x) = 0.
Students apply the test mechanically.
Fix: If f″(x) = 0 the test says nothing. Use another method.

Worked examples

Example 1

If y = 3x⁴ − 2x³ + 5x, then d²y/dx² at x = 1 is: (a) 18 (b) 24 (c) 30 (d) 36

Show the solution

First derivative: dy/dx = 12x³ − 6x² + 5.
Second derivative: d²y/dx² = 36x² − 12x.
At x = 1: 36 − 12 = 24.

Answer: (b) 24

Example 2

If y = e³ˣ, then d²y/dx² is: (a) 3e³ˣ (b) 6e³ˣ (c) 9e³ˣ (d) e³ˣ

Show the solution

First derivative: dy/dx = 3e³ˣ.
Second derivative: d²y/dx² = 3 × 3e³ˣ = 9e³ˣ.

Answer: (c) 9e³ˣ

Example 3

If y = log x (x > 0), then the value of d²y/dx² at x = 2 is: (a) −1/4 (b) 1/4 (c) −1/2 (d) 1/2

Show the solution

Write y = log x. First derivative: dy/dx = 1/x = x⁻¹.
Second derivative: d²y/dx² = −x⁻² = −1/x².
At x = 2: −1/4.

Answer: (a) −1/4

Exam tips

Questions are mostly direct. Practise polynomial, eˣ and log x types until they take under a minute.
Check the sign of the answer against the options first. It often removes two choices.
Remember the link to maxima and minima. Many questions ask for the nature of a point using f″(x).
Do not substitute the point until the last step.
If a question needs a long product or quotient rule twice, mark it and move on.

Practice questions from Differential and Integral Calculus

Higher Order Derivatives: frequently asked questions

What is a second order derivative?

It is the derivative of the first derivative. It is written f″(x) or d²y/dx². It shows how fast the slope of the function is changing.

How do I find the second derivative of a function?

Differentiate the function once and simplify. Then differentiate that result again. Use the same rules for both steps.

What is the second derivative of a linear function?

It is zero. The first derivative of ax + b is the constant a, and the derivative of a constant is zero.

Where is the second derivative used in CA Foundation?

It is used in the second derivative test for maxima and minima, and in cost and revenue problems. You study these in later topics of the same chapter.