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Business Mathematics and Statistics · Class 11 Commerce

Ch 5Differential Calculus (Functions & Graphs, Limits & Derivatives, Differentiation Techniques) — Class 11 Business Mathematics and Statistics, concept-first.

A function is a rule that assigns to every element of one set (the domain) exactly one element of another set (the codomain). In business mathematics we almost always work with real-valued functions of a real variable, written or simply .

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Key concepts

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Algebra of Limits and Standard Limits

Limits of sums, differences, constant multiples, products, and quotients (with a non-zero denominator limit) can be built from the limits of the individual pieces.

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CA Foundation

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2026, 2025

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Functions — Definition, Domain and Range

A function is a rule that assigns to every element of one set (the domain) exactly one element of another set (the codomain).

2

Types of Functions and Their Graphs

Several families of functions recur constantly in business mathematics, and it helps to recognise both their algebraic form and the shape of their graph.

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The Idea of a Limit

The whole of differential calculus rests on one idea: the limit. Informally, we ask: as gets closer and closer to some fixed value (without necessarily ever equalling ), what value does get closer and…

4

Algebra of Limits and Standard Limits

Limits of complicated expressions are almost never worked out from the informal definition directly — instead we build them from limits of simpler pieces, using the algebra of limits.

5

Evaluating Limits — Substitution and Factorisation

When we need , the first thing to try is always direct substitution: simply put into . If is a polynomial, or any 'nice' combination of standard functions defined at , this immediately gives the answe…

6

The Derivative — Definition from First Principles

Consider the graph of and two points on it, and a nearby point , where is a small (positive or negative) change in . The straight line through and is a secant (chord), and its slope is

7

Standard Derivatives Derived from First Principles

Working from the first-principles definition every single time would make calculus painfully slow, so a handful of standard results are derived once from first principles and then reused as ready-made…

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Rules of Differentiation — Sum, Difference, Product and Quotient

Real functions in business problems are built by combining simpler functions — added, subtracted, multiplied, or divided — so the next step is a set of rules for differentiating such combinations dire…

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The Chain Rule and Implicit Differentiation

Many functions met in practice are composite functions — one function applied to the output of another, such as , which is 'raise-to-the-4th' applied to ''.

Exercises

+Show 11 questions11 questions
  1. Q10Evaluate $\displaystyle\lim_{x\to2}\dfrac{x^2-4}{x-2}$.Free
  2. Q11Find the derivative of $f(x) = x^2$ from first principles.Free
  3. Q12Find the derivative of $f(x) = \dfrac{1}{x}$ from first principles.Free
  4. Q13Find the derivative of $f(x) = \sqrt{x}$ from first principles.Preview
  5. Q14Differentiate $y = x^3+5x^2-7x+2$ with respect to $x$.Preview
  6. Q15Differentiate $y = (x^2+1)(x^3-2)$ with respect to $x$, using the product rule.Preview
  7. Q16Differentiate $y = \dfrac{x+1}{x-1}$ with respect to $x$, using the quotient rule.Preview
  8. Q17Differentiate $y = (3x^2+5)^4$ with respect to $x$, using the chain rule.Preview
  9. Q18Differentiate $y = \sin(2x+1)$ with respect to $x$, using the chain rule.Preview
  10. Q19If $x^2+y^2=25$, find $\dfrac{dy}{dx}$ using implicit differentiation.Preview
  11. Q20If $xy+y^2=7$, find $\dfrac{dy}{dx}$ using implicit differentiation.Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 29 questions29 questions
  1. Q1If $f(x) = \dfrac{1-x}{1+x}$, $1+x \neq 0$, then $f(-x)$ is equal to : (a) $f(x)$ (b) $-f(x)$ (c) $\dfrac{1}{f(x)}$ (d) $-\dfrac{1}{f(x)}$Preview
  2. Q2If $y = e^{2x}$ then $y_2 = ?$ (a) $e^{-2x}$ (b) $2e^{2x}$ (c) $e^{2x}$ (d) $4e^{2x}$Preview
  3. Q3If $y = x$ and $z = \dfrac{1}{x}$, then $\dfrac{dy}{dz} =$ (a) $-\dfrac{1}{x^2}$ (b) $x^2$ (c) $1$ (d) $-x^2$Preview
  4. Q4Evaluate : $\lim_{x \to \infty} x \tan\left(\dfrac{1}{x}\right)$.Preview
  5. Q5Evaluate : $\lim_{x \to 0} \dfrac{\sqrt{2+x} - \sqrt{2-x}}{2x}$.Preview
  6. Q6Differentiate : $\dfrac{x^2+x+1}{x^2-x+1}$.Preview
  7. Q7If $f(x) = \begin{cases} x^{2}-4x & \text{if } x \geq 2 \\ x+2 & \text{if } x < 2 \end{cases}$ then, $f(0)$ is : (a) $-1$ (b) $2$ (c) $0$ (d…Preview
  8. Q8Which of the following function is neither even nor odd ? (a) $f(x)=x^{10}$ (b) $f(x)=x^{3}+5$ (c) $f(x)=x^{2}$ (d) $f(x)=x^{5}$Preview
  9. Q9If $y=x$ and $z=\dfrac{1}{x}$, then $\dfrac{dy}{dz} =$ (a) $-x^{2}$ (b) $x^{2}$ (c) $-\dfrac{1}{x^{2}}$ (d) $1$Preview
  10. Q10If $x = \dfrac{1}{t}$, $y = \cos t$ then find $\dfrac{dy}{dx}$.Preview
  11. Q11Differentiate the function $\dfrac{1 - 3x}{1 + 3x}$ with respect to $x$.Preview
  12. Q12(a) If $y = a\,\cos mx + b\,\sin mx$, then show that $y_2 + m^{2}y = 0$. OR (b) By the principle of mathematical induction prove that $n^{2}…Preview
  13. Q13The graph of $y = 2x^2$ is passing through the point : (a) $(2, 0)$ (b) $(0, 0)$ (c) $(0, 2)$ (d) $(2, 1)$Preview
  14. Q14If $f(x) = x^2 - x + 1$, then $f(x+1)$ is : (a) $1$ (b) $x^2$ (c) $x^2 + x + 1$ (d) $x$Preview
  15. Q15Evaluate : $\displaystyle\lim_{x \to \infty} \dfrac{2x + 5}{x^2 + 3x + 9}$Preview
  16. Q16Find $\dfrac{dy}{dx}$, if $x = a \sec^3\theta$, $y = b \tan^3\theta$.Preview
  17. Q17If $y=e^{2x}$, then $\dfrac{d^2y}{dx^2}$ at $x=0$ is : (a) $2$ (b) $4$ (c) $0$ (d) $9$Preview
  18. Q18If $y=4ae^{4x}$, then find $\dfrac{dy}{dx}$ : (a) $16ae^{x}$ (b) $ae^{4x}$ (c) $16ae^{4x}$ (d) $4ae^{4x}$Preview
  19. Q19If $f(x)=x^3-\dfrac{1}{x^3},\ x\neq 0$, then show that $f(x)+f\left(\dfrac{1}{x}\right)=0$.Preview
  20. Q20If $f(x)=2^x$, then show that $f(x)\cdot f(y)=f(x+y)$.Preview
  21. Q21Evaluate $\lim\limits_{x\to 0}\dfrac{\sqrt{1+x}-\sqrt{1-x}}{x}$Preview
  22. Q22$\lim\limits_{x\to 0}\dfrac{e^x-1}{x}=$ (a) $1$ (b) $e$ (c) $0$ (d) $nx^{n-1}$Preview
  23. Q23If $f(x)=\dfrac{1-x}{1+x}$, $x>1$ then $f(-x)$ is equal to : (a) $-\dfrac{1}{f(x)}$ (b) $-f(x)$ (c) $f(x)$ (d) $\dfrac{1}{f(x)}$Preview
  24. Q24Range of $\sin^{-1}x$ is ________. (a) $[0,\pi]$ (b) $\left[\dfrac{-\pi}{2},\dfrac{\pi}{2}\right]$ (c) $R$ (d) $\left(\dfrac{-\pi}{2},\dfrac…Preview
  25. Q25$\lim\limits_{\theta\to 0}\dfrac{\sin 2\theta}{2\theta}=$ ________. (a) $1$ (b) $\infty$ (c) $2$ (d) $0$Preview
  26. Q26Find the domain for which the functions $f(x)=2x^2-1$ and $g(x)=1-3x$ are equal.Preview
  27. Q27If $y=500e^{7x}+600e^{-7x}$, then show that $y_2-49y=0$.Preview
  28. Q28Evaluate : $\lim\limits_{x\to 0}\dfrac{\log\left(1+x^4\right)}{\tan^4 x}$ (Compulsory)Preview
  29. Q29(a) If $x^m \cdot y^n=(x+y)^{m+n}$, then show that $\dfrac{dy}{dx}=\dfrac{y}{x}$ OR (b) Sundar bought ₹ 4,500, 12% of ₹ 10 shares at par. He…Preview

More questions

+Show 9 questions9 questions
  1. Example 1Find the domain and range of the function $f(x) = \dfrac{1}{x-3}$.Free
  2. Example 2Find the domain and range of $f(x) = \sqrt{x-2}$.Free
  3. Example 3A function is defined by $f(x) = 2x+5$ for $x \in \mathbb{R}$. Find the value of $x$ for which $f(x)=0$, and state the domain and range of $…Free
  4. Example 4Describe the shape of the graph of $y=x^2$, stating its vertex, axis of symmetry, and the intervals where it increases and decreases.Preview
  5. Example 5Compare the graphs of $y=2^x$ and $y=\log_2 x$, stating the domain, range, and the relationship between the two graphs.Preview
  6. Example 6Evaluate $\displaystyle\lim_{x\to2}(x^2+3x-1)$.Preview
  7. Example 7Examine whether $\displaystyle\lim_{x\to0}\dfrac{|x|}{x}$ exists, by finding the left-hand and right-hand limits.Preview
  8. Example 8Evaluate $\displaystyle\lim_{x\to0}\dfrac{\sin 3x}{x}$.Preview
  9. Example 9Evaluate $\displaystyle\lim_{x\to\infty}\left(1+\dfrac{2}{x}\right)^{x}$.Preview