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Applied Mathematics · Class 12 Commerce

Ch 3Differentiation and Its Applications — Class 12 Applied Mathematics, concept-first.

This concept map shows how the chapter fits together. It has two parts: Differentiation — techniques for differentiating implicit, parametric and logarithmic functions and finding higher-order derivatives — and the Application of Derivatives — cost and revenue functions, the derivative as a rate of change (including ma…

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Implicit Differentiation

You already know how to differentiate — just apply the power rule and get . That's explicit differentiation: is written directly in terms of , so the derivative falls out cleanly.

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Chapter contents

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

Concept Map

This concept map shows how the chapter fits together. It has two parts: Differentiation — techniques for differentiating implicit, parametric and logarithmic functions and finding higher-order derivat…

3.2

Recall Some Standard Results of Differentiation

Before working through this chapter, it helps to recall the standard derivatives and differentiation rules you've already met in Class XI.

3.3

Differentiation of Implicit Functions

So far, every function you've differentiated has been given explicitly — written directly as a formula in , such as . But not every relation between and comes in that form.

+Worked Examplesi3 questions
  1. Example 1If y = f(x) is a real function, then find derivative of the following with respect to 'x'. i. $y^2$ ii. $x^3 \cdot y^5$ iii. $\log(xy^2)$ iv…Free
  2. Example 2Find $\dfrac{dy}{dx}$, when $x^3 + y^3 = xy$.Preview
  3. Example 3If $x^m \cdot y^n = (x+y)^{m+n}$, then show that $\dfrac{dy}{dx} = \dfrac{y}{x}$.Preview
3.4

Differentiation of Parametric Functions

Instead of relating and directly, it is sometimes more convenient to express both as functions of a third, auxiliary variable — a parameter, usually called .

+Worked Examplesi2 questions
  1. Example 4Find $\dfrac{dy}{dx}$ if $x = at^2$, $y = 2at$.Free
  2. Example 5Find $\dfrac{dy}{dx}\Big]_{t=1}$ if $x = \dfrac{1-t}{1+t}$, $xy = 2t^3$.Preview
3.5

Logarithmic Differentiation

The power rule applies when the exponent is a fixed number, and the exponential rule applies when the base is a fixed positive number other than 1.

+Worked Examplesi2 questions
  1. Example 6Differentiate the following with respect to 'x'. i. $y = x^x$ ii. $y = x^y$Free
  2. Example 7If $x^y + y^x = a^b$, then find $\dfrac{dy}{dx}$.Preview
3.6

Second and Higher Order Derivatives

10 Q

Since the derivative of a function is itself a function of , it can, in turn, be differentiated. Differentiating once more with respect to gives the second-order derivative, written or ; differentiati…

+Worked Examplesi2 questions
  1. Example 8Find $\dfrac{d^2y}{dx^2}$ for the following functions. i. $y = x$ ii. $y = \log x$ iii. $y = \sqrt{x^2 - 1}$Free
  2. Example 9If $x = \dfrac{t^2}{1+t}$ and $y = \dfrac{t}{1+t}$, find $y_2$.Preview
+3.18 questions
  1. Q1Find $\dfrac{dy}{dx}$ from the following i. $x^3 + y^3 = 3axy$ ii. $e^{xy} - axy = a$ iii. $3x^3 - 5x^2y + 2xy^2 + 4y^3 = 0$ iv. $x^{1/3} +…Free
  2. Q2Find $\dfrac{dy}{dx}$ from the following parametric equations i. $x = at$, $y = \dfrac{a}{t}$ ii. $x = t\cdot\log t$, $y = \dfrac{\log t}{t}…Free
  3. Q3Find $\dfrac{dy}{dx}$ from the following equations i. $x^y = y^x$ ii. $x^y = e^{x-y}$ iii. $(x - y)e^{x/(x-y)} = 7$ iv. $y = x^{\log x}$Free
  4. Q4Find $\dfrac{d^2y}{dx^2}$ from the following (i) $y = x\log x$ (ii) $y = x^2 e^x$ (iii) $y = \log(\log x)$ (iv) $y = 3e^{2x} + 2e^{3x}$Preview
  5. Q5If $x\sqrt{1+y} + y\sqrt{1+x} = 0$, show that $(1+x^2)\dfrac{dy}{dx} + 1 = 0$.Preview
  6. Q6If $y^{1/m} + y^{-1/m} = 2x$ then prove that $(x^2 - 1)y_1^2 = m^2 y^2$.Preview
  7. Q7If $y = \log\left(x + \sqrt{a^2 + x^2}\right)$, show that $(a^2 + x^2)y_2 + xy_1 = 0$.Preview
  8. Q8If $y = \left(x + \sqrt{x^2 + 1}\right)^p$, prove that $(x^2 + 1)y_2 + xy_1 - p^2 y = 0$.Preview
3.7

Cost and Revenue Function

Every manufacturing business incurs two distinct kinds of cost. Variable cost covers expenses like raw material, direct labour, and packaging — it moves up or down with the level of production, rising…

+Worked Examplesi1 question
  1. Example 10A company produces 'x' units in a year and the variable cost is $V(x) = x^2 - 2x$. Also, the company spends a fixed cost of Rs 15,000 on com…Preview
3.8

Derivative as Rate of Change of Quantities

Many real situations involve one quantity changing because another one does — distance changes with time, cost changes with production, revenue changes with production, price changes with demand.

+Worked Examplesi1 question
  1. Example 11A boy is blowing air into a spherical balloon and thus the radius r of the balloon is changing, then find the rate of change of surface area…Preview
3.8.1

Related Rates

In a related-rate problem, two or more quantities are changing simultaneously with respect to time, and you are given the rate of change of one of them in order to find the rate of change of another.

+Worked Examplesi3 questions
  1. Example 12Find the rate of change of volume of a sphere with respect to its surface area when the radius is 5 m.Free
  2. Example 13A cylindrical vessel of radius 0.5 m is filled with oil at the rate of $0.25\pi \ \text{m}^3/\text{min}$. Find the rate at which the surface…Preview
  3. Example 14A boy of height 1 m is walking towards a lamp post of height 5 meters at the rate of 0.5 m/sec. Then find the rate at which the length of th…Preview
3.8.2

Marginal Cost and Marginal Revenue

12 Q

Marginal cost and marginal revenue are just the rate-of-change idea from the previous section applied to a firm's cost and revenue functions.

+Worked Examplesi2 questions
  1. Example 15A toy manufacturing firm assesses its variable cost to be 'x' times the sum of 30 and 'x', where 'x' is the number of toys produced, also th…Free
  2. Example 16The price per unit of a commodity produced by a company is given by $p = 30 - 2x$ and 'x' is the quantity demanded. Find the revenue functio…Preview
+3.210 questions
  1. Q1Find the rate of change of circumference of a circle with respect to the radius r.Free
  2. Q2Find the rate of change of lateral surface area of a cube with respect to side x, when x = 4 cm.Free
  3. Q3If the rate of change of volume of a sphere is equal to the rate of change of its radius, then find its radius. Also find its surface area.Free
  4. Q4The volume of a cone changes at the rate 40 cm³/sec. If height of the cone is always equal to its diameter, then find the rate of change of…Preview
  5. Q5For what values of x is the rate of increase of total cost function $C(x) = x^3 - 5x^2 + 5x + 8$ is twice the rate of increase of x?Preview
  6. Q6The radius of the base of a cone is increasing at the rate of 3 cm/minute and the altitude is decreasing at the rate of 4 cm/minute. Find th…Preview
  7. Q7A ladder 10 meters long rests with one end against a vertical wall, the other on the floor. The lower end moves away from the wall at the ra…Preview
  8. Q8A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³ /min. When the thick…Preview
  9. Q9A stationery company manufactures 'x' units of pen in a given time, if the cost of raw material is square of the pens produced, cost of tran…Preview
  10. Q10A firm knows that the price per unit 'p' for one of its product is linear. It also knows that it can sell 1400 units when the price is ₹4 pe…Preview
3.9

Slope (or Gradient) of Tangent and Normal

11 Q

Take a curve , a point on it, and a nearby point . The straight line joining and is a secant, and its slope is , where is the secant's angle of inclination.

+Worked Examplesi1 question
  1. Example 17Find slope of the tangent and normal at a point (2, 6) to the curve $y = x^3 - x$.Preview
+3.310 questions
  1. Q1Find the slopes of the tangents and normal to the curves at the indicated points. i. $y = x^3 - x$ at $x = 1$. ii. $y = 3x^2 - 6x$ at $x = 2…Free
  2. Q2Find the equations of the tangent and normal to the curves at the indicated points. i. $y = x^3 - 3x + 5$ at the point (2, 7). ii. $x = at^2…Free
  3. Q3Find the equations of the tangents to the curve at points where the tangents to the curve $y = 2x^3 - 15x^2 + 36x - 21$ are parallel to x-ax…Free
  4. Q4Find the equation of the tangents to the curve $y = x^3 + 2x - 4$, which is perpendicular to the line $x + 14y + 3 = 0$.Preview
  5. Q5Find the equation of the tangent and the normal to the curve $y = \dfrac{x-7}{x^2 - 5x + 6}$ at the point, where it cuts x-axis.Preview
  6. Q6Find the equation of the normal to the curve $x^2 = 4y$ which passes through the point (1, 2).Preview
  7. Q7For the curve $y = x^2 + 3x + 4$, find all points at which the tangent passes through the origin.Preview
  8. Q8Show that the line $\dfrac{x}{a} + \dfrac{y}{b} = 1$ touches the curve $y = be^{-x/a}$ at the point where it crosses the y-axis.Preview
  9. Q9Show that the curves $xy = a^2$ and $x^2 + y^2 = 2a^2$ touch each other.Preview
  10. Q10Prove that the curves $xy = 4$ and $x^2 + y^2 = 8$ touch each other.Preview
3.9.1

Equation of Tangent and Normal to a Curve

Once you know the slope of the tangent (or normal) at a point on a curve, writing down its equation is just an application of the familiar point-slope form: a line through with slope satisfies .

+Worked Examplesi7 questions
  1. Example 18Find the equation of the tangent and normal to the curve $x^{2/3} + y^{2/3} = 2$ at (1, 1).Free
  2. Example 19Find the equation of the tangent and normal to the curve $f(x) = e^x + x^2 + 1$ at the point (0, 2) on it.Free
  3. Example 20Find the equation of the tangent to the curve $x^2 + 3y - 3 = 0$, which is parallel to the line $y = 4x - 5$.Free
  4. Example 21Find the equation of the tangent to the curve $y = (x^3 - 1)(x - 2)$ at the points where the curve cuts the x-axis.Preview
  5. Example 22Find the equation of the tangents to the curve $3x^2 - y^2 = 8$, which passes through the point $\left(\dfrac{4}{3}, 0\right)$.Preview
  6. Example 23Prove that $\left(\dfrac{x}{a}\right)^n + \left(\dfrac{y}{b}\right)^n = 2$ touches the straight line $\dfrac{x}{a} + \dfrac{y}{b} = 2$ for a…Preview
  7. Example 24Find the points on the curve $y = 4x^3 - 2x^5$, at which the tangent passes through the origin.Preview
3.10

Increasing and Decreasing Functions (Monotonic Functions)

9 Q

A function's derivative doesn't just give the slope at a point — it also tells you whether the function is climbing or falling as increases, which is the idea of monotonicity.

+Worked Examplesi1 question
  1. Example 25Show that $f(x) = x^2$ is an (i) Increasing function on $(0, \infty)$ (ii) Decreasing function on $(-\infty, 0)$Preview
+3.48 questions
  1. Q1Find critical points of the following functions i. $f(x) = x^3 - 6x^2 + 9x - 10$ ii. $f(x) = \dfrac{\log x}{x},\ x > 0$ iii. $f(x) = 50\sqrt…Free
  2. Q2Find the intervals in which the following functions are increasing or decreasing i. $f(x) = x^4 - 8x^3 + 22x^2 - 24x + 1$ ii. $f(x) = (x + 2…Free
  3. Q3Show that the function $f(x) = \log(1 + x) + \dfrac{1}{1 + x}$ increases on $(0, \infty)$.Free
  4. Q4Prove that the function $f(x) = x^2 - x + 1$ is neither increasing nor decreasing in (0, 1).Preview
  5. Q5A company finds that its total revenue may be determined by $R(x) = \left[240000 - (x - 500)^2\right]$. Find when is the revenue function in…Preview
  6. Q6The price 'p' per unit is given by the relation $x = \dfrac{1}{3}p^2 - 2p + 3$ where 'x' is the number of units sold then, i. Find the reven…Preview
  7. Q7The total cost function of a manufacturing company is given by $C(x) = 2x\left(\dfrac{x+4}{x+3}\right) + 3$. Show that MC (Marginal Cost) fa…Preview
  8. Q8The price 'p' per unit at which a company can sell all that it produces is given by p = 29 – x, where 'x' is the number of units produced. T…Preview
3.10.1

Derivative Conditions for a Monotonic (Increasing or Decreasing) Function

The link between a function's derivative and its monotonicity follows from the geometric meaning of the derivative established earlier in this chapter: at any point, is the slope of the tangent there.…

+Worked Examplesi4 questions
  1. Example 26Find the critical point(s) of the following functions i. $f(x) = 12x^{4/3} - 6x^{1/3}$ on $[-1, 1]$ ii. $f(x) = \dfrac{x^4}{4} - 2x^3 + \dfr…Free
  2. Example 27Using derivatives check whether the following functions are monotonic i. $f(x) = x^2$ on $(0, \infty)$ ii. $f(x) = x^2$ on $(-\infty, 0)$ ii…Free
  3. Example 28Find the intervals in which the function $f(x) = \dfrac{x^4}{4} - 2x^3 + \dfrac{11}{2}x^2 - 6x$ is (i) Increasing (ii) Decreasing.Preview
  4. Example 29The cost function C(x) of a commodity is given as $C(x) = 2x\left(\dfrac{x+3}{x+2}\right) + 2$. Prove that the marginal cost falls as the ou…Preview
3.11

Maxima and Minima

20 Q

Beyond knowing where a function rises or falls, it is often just as important to know where it reaches its highest or lowest value overall — its extreme values.

+Worked Examplesi12 questions
  1. Example 30Find the maximum (absolute) and minimum (absolute) value of the following functions. i. $f(x) = |x| + 3$ ii. $f(x) = 9x^2 + 12x + 2$ iii. $f…Free
  2. Example 31Find all the points of local maxima and local minima and the local maximum and local minimum values of the function $f(x) = x^4 - 8x^3 + 22x…Free
  3. Example 32Use the second derivative test to find the local maxima and minima of $f(x) = \dfrac{4}{3}x^3 + 6x^2 + 8x + 7$.Free
  4. Example 33Use the second derivative test to find the local maxima and minima of $f(x) = x^3 - 3x^2 + 3x + 5$, if any.Preview
  5. Example 34Find the absolute maximum and minimum value of the function $f(x) = x^3 - \dfrac{3}{2}x^2 - 18x + 1$ on $[-4, 6]$.Preview
  6. Example 35Find two positive numbers whose sum is 16 and whose product is as large as possible.Preview
  7. Example 36The production manager of a company plans to include 180 square centimetres of actual printed matter in each page of a book under production…Preview
  8. Example 37An open tank with a square bottom is to contain 4000 cubic cm of liquid is to be constructed. Find the dimension of the tank so that the sur…Preview
  9. Example 38A wire 40 m length is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be t…Preview
  10. Example 39Let the cost function of firm be given by the equation $C(x) = 300x - 10x^2 + \dfrac{1}{3}x^3$. Find the output at which the marginal cost M…Preview
  11. Example 40A pen drive manufacturing company charges ₹6,000 per unit for an order of 50 pen drives or less. The charge is reduced by ₹75 per pen drive…Preview
  12. Example 41A manufacturer produces x pants per week at total cost of ₹$(x^2 + 78x + 2500)$. The price per unit is given by $8x = 600 - p$, where 'p' is…Preview
+3.58 questions
  1. Q1Find the local maxima, local minima, local minimum value and local maximum value, if any of the following i. $f(x) = x^2 - 6x + 16$ ii. $f(x…Free
  2. Q2The sum of two positive numbers is 16. Find the numbers, if the product of the squares is to be maximum.Free
  3. Q3Show that of all rectangles with a given perimeter, the square has the largest area.Free
  4. Q4Show that the function $f(x) = x^3 - 6x^2 + 12x + 50$ has neither a local maximum nor a local minimum value.Preview
  5. Q5The profit function, in rupees, of a firm selling 'x' items $(x \geq 0)$ per week is given by P(x) = (400 – x)x – 3500. How many items shoul…Preview
  6. Q6A tour operator charges ₹136 per passenger for 100 passengers with a discount of ₹4 for each 10 passengers in excess of 100. Find the number…Preview
  7. Q7If price 'p' per unit of an article is p = 75 – 2x and the cost function is $C(x) = 350 + 12x + \dfrac{x^2}{4}$. Find the number of units an…Preview
  8. Q8The cost of fuel in running an engine is proportional to the square of the speed in kms per hour, and is ₹48 per hour when the speed is 16 k…Preview

Summary

This chapter developed the techniques of differentiation and applied them to business and geometric problems. Its key ideas and formulae are collected below.

Case-Based Questions

+Case Based Questions2 questions
  1. Q1**Case Study-I.** A farmer has a piece of land. He observed that he got 600 units of fruits per tree by planting upto 25 trees and when 26 t…Free
  2. Q2**Case Study-II.** A manufacturing company manufactures toys; the company observed the following costs at different production levels: | Num…Preview