Skip to content
← Statistics

Statistics · Class 12 Commerce

Ch 9Differentiation — Class 12 Statistics, concept-first.

Differentiation is the branch of mathematics that studies how one quantity changes in response to a change in another — its rate of change. For a Gujarat Board (GSHSEB) Std 12 Commerce Statistics student, this idea matters directly: cost, revenue, profit and demand are all quantities that change as output, price, or ti…

43

Q&A

5

Concepts

Not available

Exam weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Meaning of the Derivative and Rate of Change

The derivative of with respect to , written or , is the limit of the average rate of change as : . It measures the instantaneous rate of change of with respect to , and geometrically equals the slope of the tangent to th…

Most relevant Q&A

Start with this concept →

Chapter contents

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

1

Meaning of the Derivative and Rate of Change

Differentiation is the branch of mathematics that studies how one quantity changes in response to a change in another — its rate of change.

2

Rules of Differentiation

Differentiating every function from first principles, using limits, would be slow and error-prone. Instead, a small set of standard rules of differentiation — each itself provable from first principle…

3

Derivatives of Standard Functions

Beyond powers of , Gujarat board class 12 commerce statistics problems (particularly business and growth applications) also use exponential and logarithmic functions.

4

Marginal Cost and Marginal Revenue

Differentiation's most direct use in commerce is measuring how cost and revenue change as the level of output changes — a topic that is a recurring feature of Gujarat board class 12 commerce statistic…

5

Maxima and Minima (Profit Maximisation)

A central business application of differentiation is finding the level of output at which a quantity — most importantly, profit — is at its maximum (or, for cost, at its minimum).

Exercises

+Show 6 questions6 questions
  1. Q9Differentiate $f(x) = 2x^2 + 3x$ from first principles.Free
  2. Q10Differentiate $y = 7x^5 - 2x^3 + 4x - 1$ with respect to $x$.Free
  3. Q11Using the product rule, differentiate $y=(3x-2)(x^2+4)$.Preview
  4. Q12Differentiate $y = 5e^x + 2\ln(3x)$ with respect to $x$.Preview
  5. Q13The demand function for a product is $P = 200 - 4Q$. Find the Total Revenue and Marginal Revenue functions, and the Marginal Revenue at $Q=2…Preview
  6. Q14Total Revenue is $TR = 100Q - 2Q^2$ and Total Cost is $TC = Q^2 + 20Q + 50$. Find the output level $Q$ that maximises profit, and confirm it…Preview

Sample & Board Papers

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

+Show 29 questions29 questions
  1. Q1If $y = ax + b$, where $a$ and $b$ are constants then what will be $\frac{dy}{dx}$? (a) $a$ (b) $b$ (c) $a + b$ (d) $0$Preview
  2. Q2State the formula of elasticity of demand.Preview
  3. Q3Find $\dfrac{dy}{dx}$ for $y = x^3 + \sqrt{x} - \dfrac{4}{x} + \dfrac{1}{\sqrt[3]{x}} + \dfrac{1}{4}$.Preview
  4. Q4If $f(x) = 4x^3 + 2x^2 + 7x + 9$ then for which value of $x$ is $f''(x) = 52$?Preview
  5. Q5Find the maximum and minimum values of $y = x^3 - 2x^2 - 4x - 1$. OR The daily cost of production for $x$ tons of a commodity is $10x^2 - 10…Preview
  6. Q6If $y = ax + b$, $a$ and $b$ are constant then what will be $\dfrac{dy}{dx}$? (a) $a$ (b) $b$ (c) $a + b$ (d) 0Preview
  7. Q7If $u$ and $v$ are two functions of $x$ then what is the formula of derivative of their product? (a) $u\dfrac{du}{dx} + v\dfrac{dv}{dx}$ (b)…Preview
  8. Q8Find $\dfrac{dy}{dx}$ if $y = 6x^3 + \dfrac{7}{2}x^2 + \dfrac{6}{5}x - 8$.Preview
  9. Q9Find $f''(0)$, if $f(x) = x^4 - 4x^3 + 3x^2 + x + 1$.Preview
  10. Q10If $y = \dfrac{2x^2 + 3x + 4}{x^2 + 5}$ the find $\dfrac{dy}{dx}$.Preview
  11. Q11A producer produces $x$ units at cost $200x + 15x^2$. The demand function is $P = 1200 - 10x$. Find the profit function and how many units s…Preview
  12. Q12If $u$ and $v$ are two functions of $x$ then what is the formula of derivative of their product? (a) (A) $u\frac{du}{dx} + v\frac{dv}{dx}$ (…Preview
  13. Q13Find $f'(x)$ if $f(x) = 9x^2 - 8x + 6$.Preview
  14. Q14Determine whether the function $y = 3 + 2x - 7x^2$ is increasing or decreasing at $x = -4$ and $x = 4$.Preview
  15. Q15If $f(x) = 4x^3 + 2x^2 + 7x + 9$ then for which value of $x$, $f''(x) = 52$?Preview
  16. Q16Find the maximum and minimum values of $y = x^3 - 2x^2 - 4x - 1$. OR The selling price of a refrigerator as determined by the company is ₹10…Preview
  17. Q17What is $\frac{dy}{dx}$ if $y = ax^n$, $a$ is constant? (a) $nx^{n-1}$ (b) $anx^{n-1}$ (c) $0$ (d) $anx^{n+1}$Preview
  18. Q18What is the formula for elasticity of demand? (a) $-\frac{P}{x} \cdot \frac{dx}{dp}$ (b) $\frac{P}{x} \cdot \frac{dx}{dp}$ (c) $-\frac{x}{p}…Preview
  19. Q19Define marginal cost.Preview
  20. Q20Determine whether the function $y = 12 + 4x - 7x^2$ is increasing or decreasing at $x = 2$.Preview
  21. Q21Find $f'(x)$ if $f(x) = (x^2 + 3x + 4)^7$.Preview
  22. Q22The demand function of an item is $P = 30 - \frac{x^2}{10}$. Find the demand and price for maximum revenue.Preview
  23. Q23If $y = ax + b$, $a$ and $b$ are constant then what will be $\dfrac{dy}{dx}$? (a) $a$ (b) $b$ (c) $a + b$ (d) $0$Preview
  24. Q24If the function $f(x)$ is increasing at $x = a$ then which is the correct option from the following? (a) $f'(a) < 0$ (b) $f'(a) > 0$ (c) $f'…Preview
  25. Q25How will be the first order derivative of a function at $x = a$ if function is decreasing at $x = a$?Preview
  26. Q26If $f(x) = 3x^2 + 2x + 1$ then find $f'(x)$ and hence obtain $f'(-1)$.Preview
  27. Q27If the demand function of pizza is $p = 150 - 4x$ then find the marginal revenue when demand is of 3 pizzas.Preview
  28. Q28Find the value of $\lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}$ where $f(x) = \sqrt{x},\ x > 0$.Preview
  29. Q29The daily cost of production for $x$ tons of a commodity is $10x^2 - 1000x + 50{,}000$. How many units should be produced for the minimum co…Preview

More questions

+Show 8 questions8 questions
  1. Example 1Differentiate $f(x) = x^3$ from first principles.Free
  2. Example 2Differentiate $y = 4x^3 - 6x^2 + 5x - 9$ with respect to $x$.Free
  3. Example 3Using the product rule, differentiate $y = (x^2+3)(2x-1)$.Free
  4. Example 4Using the quotient rule, differentiate $y = \dfrac{x+2}{x^2+1}$.Preview
  5. Example 5Using the chain rule, differentiate $y = (2x^2+3x)^5$.Preview
  6. Example 6Differentiate $y = 3e^{2x} - 4\ln x$ with respect to $x$.Preview
  7. Example 7The total cost of producing $Q$ units is $TC = 2Q^2 + 10Q + 50$ (₹). Find the Marginal Cost function and the Marginal Cost at $Q=5$ units.Preview
  8. Example 8The total cost function for a firm is $TC = Q^2 - 20Q + 500$. Find the output level that minimises total cost, and confirm it is genuinely a…Preview