Business Mathematics and Statistics · Class 12 Commerce
Ch 6Differentiation — Class 12 Business Mathematics and Statistics, concept-first.
For an Odisha CHSE +2 Commerce student, differentiation is the single mathematical idea used most often to explain how a business quantity — cost, revenue, profit, or price — behaves as another quantity (usually output or time) changes.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Derivative as a Rate of Change — First Principles
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 the…
Most relevant Q&A
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Derivative as a Rate of Change — First Principles
For an Odisha CHSE +2 Commerce student, differentiation is the single mathematical idea used most often to explain how a business quantity — cost, revenue, profit, or price — behaves as another quanti…
Standard Rules of Differentiation
Differentiating every function from the limit definition would be slow and error-prone. A small set of standard rules of differentiation — each of them itself provable from first principles — lets alm…
Derivatives of Exponential and Logarithmic Functions
Beyond powers of , business and growth problems in this syllabus regularly use the exponential function and the natural logarithm.
Higher-Order Derivatives
Differentiating a function once gives its first derivative, or . Differentiating that result again, with respect to once more, gives the second derivative, written , , or .
Marginal Cost and Marginal Revenue
One of the most direct commerce applications of differentiation is measuring how total cost and total revenue change as the level of output changes.
Maxima, Minima and Profit Maximisation
A central business application of differentiation is finding the output level at which a quantity — most importantly, profit — is at its maximum (or, for cost, at its minimum).
Exercises
+−Show 4 questionsHide questions4 questions
- Q11Differentiate $f(x) = 3x^2 + 2x$ from first principles.Free
- Q12(MCQ) The derivative of $\ln(5x)$ with respect to $x$ is: (a) $\dfrac{1}{x}$ (b) $\dfrac{5}{x}$ (c) $\dfrac{\ln5}{x}$ (d) $5$Free
- Q13(MCQ) If the total cost function is $TC = Q^2 + 40$, the Marginal Cost at $Q=6$ is: (a) $6$ (b) $12$ (c) $46$ (d) $76$Preview
- Q14The total cost function for a firm is $TC = Q^2 - 16Q + 200$. Find the output level that minimises total cost, and confirm it is genuinely a…Preview
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 10 questionsHide questions10 questions
- Q1Answer each of the following questions in one sentence: (xi) Differentiate $3r^2 + 4r + 7$ with respect to $r$.Preview
- Q2(j) The derivative of $\sqrt{x}$ with respect to $x$, is (a) $\dfrac{1}{2}\sqrt{x}$ (b) $\dfrac{1}{2} x^{3/2}$ (c) $\dfrac{1}{2} x\sqrt{x}$…Preview
- Q3(d) If $y = (2x^3 - 1)^4$, find $\dfrac{dy}{dx}$.Preview
- Q4(d) Find the differential coefficient of $x \cdot \log_e x$.Preview
- Q5$\dfrac{d}{dx}(5x^6)$ is equal to : (a) $x^{30}$ (b) $5x^5$ (c) $30x^5$ (d) $30x$Preview
- Q6Find the derivative of the following function with respect to $x$. $y = x^5 + 2x - x^3$Preview
- Q7The derivative of $5x^4 - 3x^3 + 2x^2 - 11x + 7$ with respect to x, is : (a) $20x^3 + 9x^2 - 4x + 11$ (b) $20x^3 - 9x^2 + 4x - 11$ (c) $20x^…Preview
- Q8Answer each of the following question in one sentence each : Write the formula for determining the derivative of the product of two function…Preview
- Q9Differentiate $\left(x + \frac{1}{x}\right)^2$ with respect to x.Preview
- Q10Differentiate $\sqrt{\frac{1+x}{1-x}}$ with respect to x.Preview
More questions
+−Show 10 questionsHide questions10 questions
- Example 1Differentiate $f(x) = x^3 - x$ from first principles.Free
- Example 2Differentiate $y = 5x^4 - 3x^3 + 2x^2 - 7x + 6$ with respect to $x$.Free
- Example 3Using the product rule, differentiate $y = (3x+1)(x^2-2)$.Free
- Example 4Using the quotient rule, differentiate $y = \dfrac{2x-3}{x^2+4}$.Preview
- Example 5Using the chain rule, differentiate $y = (3x^2 - 5x)^4$.Preview
- Example 6Differentiate $y = 2e^{5x} - 3\ln x$ with respect to $x$.Preview
- Example 7Find the first and second derivatives of $y = 2x^5 - 3x^3 + 4x$.Preview
- Example 8The total cost of producing $Q$ units is $TC = 3Q^2 + 6Q + 30$ (Rs). Find the Marginal Cost function and the Marginal Cost at $Q=8$ units.Preview
- Example 9The demand function for a product is $P = 150 - 3Q$. Find the Total Revenue and Marginal Revenue functions, and the Marginal Revenue at $Q=1…Preview
- Example 10Total Revenue is $TR = 60Q - 2Q^2$ and Total Cost is $TC = Q^2 + 8Q + 40$. Find the output level $Q$ that maximises profit, and confirm it i…Preview