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Mathematics · Class 12 Science

Ch 8Differentials and Partial Derivatives — Class 12 Mathematics, concept-first.

"He who hasn't tasted bitter things hasn't earned sweet things" — Gottfried Wilhelm Leibniz (1646–1716).

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Linear Approximation and Differentials

A nonlinear function is generally hard to evaluate exactly, but near any one point its graph looks almost like a straight line -- the tangent line at that point. This is the idea behind linear approximation.

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8.1

Introduction

"He who hasn't tasted bitter things hasn't earned sweet things" — Gottfried Wilhelm Leibniz (1646–1716).

8.2

Linear Approximation and Differentials

This section builds the chapter's two central tools for a function of one variable: the linear approximation (the tangent-line formula that lets us estimate near a chosen point without computing exact…

8.2.1

Linear Approximation

Definition 8.1 (Linear Approximation). Let be differentiable and . The linear approximation of at is is precisely the tangent line to at the point : it passes through that point (since ) with slope .

8.2.2

Errors: Absolute Error, Relative Error, and Percentage Error

When a value is only approximated — never computed exactly — an honest report of the approximation must also say how far off it might be.

+Exercise 8.1i7 questions
  1. Q1Let $f(x)=\sqrt[3]{x}$. Find the linear approximation at $x=27$. Use the linear approximation to approximate $\sqrt[3]{27.2}$.Free
  2. Q2Use the linear approximation to find approximate values of (i) $(123)^{2/3}$ (ii) $\sqrt[4]{15}$ (iii) $\sqrt[3]{26}$Free
  3. Q3Find a linear approximation for the following functions at the indicated points. (i) $f(x)=x^3-5x+12,\ x_0=2$ (ii) $g(x)=\sqrt{x^2+9},\ x_0=…Free
  4. Q4The radius of a circular plate is measured as $12.65$ cm instead of the actual length $12.5$ cm. Find the following in calculating the area…Preview
  5. Q5A sphere is made of ice having radius $10$ cm. Its radius decreases from $10$ cm to $9.8$ cm. Find approximations for the following: (i) cha…Preview
  6. Q6The time $T$, taken for a complete oscillation of a single pendulum with length $l$, is given by the equation $T=2\pi\sqrt{\dfrac{l}{g}}$, w…Preview
  7. Q7Show that the percentage error in the $n$th root of a number is approximately $\dfrac1n$ times the percentage error in the number.Preview
8.2.3

Differentials

Returning to the derivative (Leibniz's notation for the differential coefficient): is it meaningful to treat as an actual quotient of two separate quantities and — not merely a single symbol for a lim…

+Exercise 8.2i11 questions
  1. Q1Find differential $dy$ for each of the following functions: (i) $y=\dfrac{(1-2x)^3}{3-4x}$ (ii) $y=(3+\sin(2x))^{2/3}$ (iii) $y=e^{x^2-5x+7}…Free
  2. Q2Find $df$ for $f(x)=x^2+3x$ and evaluate it for (i) $x=2$ and $dx=0.1$ (ii) $x=3$ and $dx=0.02$Free
  3. Q3Find $\Delta f$ and $df$ for the function $f$ for the indicated values of $x,\Delta x$ and compare. (i) $f(x)=x^3-2x^2;\ x=2,\ \Delta x=dx=0…Free
  4. Q4Assuming $\log_{10}e=0.4343$, find an approximate value of $\log_{10}1003$.Preview
  5. Q5The trunk of a tree has diameter $30$ cm. During the following year, the circumference grew $6$ cm. (i) Approximately, how much did the tree…Preview
  6. Q6An egg of a particular bird is very nearly spherical. If the radius to the inside of the shell is $5$ mm and radius to the outside of the sh…Preview
  7. Q7Assume that the cross section of the artery of human is circular. A drug is given to a patient to dilate his arteries. If the radius of an a…Preview
  8. Q8In a newly developed city, it is estimated that the voting population (in thousands) will increase according to $V(t)=30+12t^2-t^3,\ 0\le t\…Preview
  9. Q9The relation between the number of words $y$ a person learns in $x$ hours is given by $y=52\sqrt x,\ 0\le x\le 9$. What is the approximate n…Preview
  10. Q10A circular plate expands uniformly under the influence of heat. If it's radius increases from $10.5$ cm to $10.75$ cm, then find an approxim…Preview
  11. Q11A coat of paint of thickness $0.2$ cm is applied to the faces of a cube whose edge is $10$ cm. Use the differentials to find approximately h…Preview
8.3

Functions of Several Variables

A function of one variable is understood through the curve in the -plane. A function of two variables is understood the same way, one dimension up: graph in -space.

8.3.1

Recall of Limit and Continuity of Functions of One Variable

Before extending limits and continuity to two variables, it helps to restate the one-variable definitions (from Class XI) in the language of neighbourhoods, since that is the form that generalizes cle…

8.4

Limit and Continuity of Functions of Two Variables

Definition 8.6 (Limit of a Function of Two Variables). Let and . has a limit at if: for every neighbourhood , , of , there exists a -neighbourhood of such that We write if such a limit exists.

+Exercise 8.3i7 questions
  1. Q1Evaluate $\displaystyle\lim_{(x,y)\to(1,2)}g(x,y)$, if the limit exists, where $g(x,y)=\dfrac{3x^2-xy}{x^2+y^2+3}$.Free
  2. Q2Evaluate $\displaystyle\lim_{(x,y)\to(0,0)}\cos\left(\dfrac{x^3+y^2}{x+y+2}\right)$, if the limit exists.Free
  3. Q3Let $f(x,y)=\dfrac{y^2-xy}{\sqrt x-\sqrt y}$ for $(x,y)\ne(0,0)$. Show that $\displaystyle\lim_{(x,y)\to(0,0)}f(x,y)=0$.Free
  4. Q4Evaluate $\displaystyle\lim_{(x,y)\to(0,0)}\cos\left(\dfrac{e^x\sin y}{y}\right)$, if the limit exists.Preview
  5. Q5Let $g(x,y)=\dfrac{x^2y}{x^4+y^2}$ for $(x,y)\ne(0,0)$ and $f(0,0)=0$. (i) Show that $\displaystyle\lim_{(x,y)\to(0,0)}g(x,y)=0$ along every…Preview
  6. Q6Show that $f(x,y)=\dfrac{x^2-y^2}{y^2+1}$ is continuous at every $(x,y)\in\mathbb R^2$.Preview
  7. Q7Let $g(x,y)=\dfrac{e^y\sin x}{x}$, for $x\ne0$ and $g(0,0)=1$. Show that $g$ is continuous at $(0,0)$.Preview
8.5

Partial Derivatives

Motivation. For , holding fixed turns into a function of alone, whose graph is the curve cut from the surface by the plane ; its ordinary derivative with respect to , evaluated at , is the slope of th…

+Exercise 8.4i10 questions
  1. Q1Find the partial derivatives of the following functions at the indicated points. (i) $f(x,y)=3x^2-2xy+y^2+5x+2,\quad (2,-5)$ (ii) $g(x,y)=3x…Free
  2. Q2For each of the following functions find the $f_x,f_y$, and show that $f_{xy}=f_{yx}$. (i) $f(x,y)=\dfrac{3x}{y+\sin x}$ (ii) $f(x,y)=\tan^{…Free
  3. Q3If $U(x,y,z)=\dfrac{x^2+y^2}{xy}+3z^2y$, find $\dfrac{\partial U}{\partial x},\dfrac{\partial U}{\partial y}$, and $\dfrac{\partial U}{\part…Free
  4. Q4If $U(x,y,z)=\log(x^3+y^3+z^3)$, find $\dfrac{\partial U}{\partial x}+\dfrac{\partial U}{\partial y}+\dfrac{\partial U}{\partial z}$.Preview
  5. Q5For each of the following functions find the $g_{xy},g_{xx},g_{yy}$ and $g_{yx}$. (i) $g(x,y)=xe^y+3x^2y$ (ii) $g(x,y)=\log(5x+3y)$ (iii) $g…Preview
  6. Q6Let $w(x,y,z)=\dfrac{1}{\sqrt{x^2+y^2+z^2}},\ (x,y,z)\ne(0,0,0)$. Show that $\dfrac{\partial^2 w}{\partial x^2}+\dfrac{\partial^2 w}{\partia…Preview
  7. Q7If $V(x,y)=e^x(x\cos y-y\sin y)$, then prove that $\dfrac{\partial^2 V}{\partial x^2}+\dfrac{\partial^2 V}{\partial y^2}=0$.Preview
  8. Q8If $w(x,y)=xy+\sin(xy)$, then prove that $\dfrac{\partial^2 w}{\partial y\,\partial x}=\dfrac{\partial^2 w}{\partial x\,\partial y}$.Preview
  9. Q9If $v(x,y,z)=x^3+y^3+z^3+3xyz$, show that $\dfrac{\partial^2 v}{\partial y\,\partial z}=\dfrac{\partial^2 v}{\partial z\,\partial y}$.Preview
  10. Q10A firm produces two types of calculators each week, $x$ number of type $A$ and $y$ number of type $B$. The weekly revenue and cost functions…Preview
8.6

Linear Approximation and Differential of a Function of Several Variables

Just as a differentiable one-variable function is well approximated near by its tangent line, a function of two (or three) variables is well approximated near a point by its tangent plane (respectivel…

+Exercise 8.5i4 questions
  1. Q1If $w(x,y)=x^3-3xy+2y^2,\ x,y\in\mathbb R$, find the linear approximation for $w$ at $(1,-1)$.Free
  2. Q2Let $z(x,y)=x^2y+3xy^4,\ x,y\in\mathbb R$. Find the linear approximation for $z$ at $(2,-1)$.Free
  3. Q3If $v(x,y)=x^2-xy+\dfrac14y^2+7,\ x,y\in\mathbb R$, find the differential $dv$.Preview
  4. Q4Let $V(x,y,z)=xy+yz+zx,\ x,y,z\in\mathbb R$. Find the differential $dV$.Preview
8.6.1

Function of Function Rule

When the two variables of are themselves each functions of a single variable (with the same domain), the composite ultimately depends only on — so it should be treatable as an ordinary one-variable fu…

+Exercise 8.6i9 questions
  1. Q1If $u(x,y)=x^2y+3xy^4,\ x=e^t$ and $y=\sin t$, find $\dfrac{du}{dt}$ and evaluate it at $t=0$.Free
  2. Q2If $u(x,y,z)=xy^2z^3,\ x=\sin t,\ y=\cos t,\ z=1+e^{2t}$, find $\dfrac{du}{dt}$.Free
  3. Q3If $w(x,y,z)=x^2+y^2+z^2,\ x=e^t,\ y=e^t\sin t$ and $z=e^t\cos t$, find $\dfrac{dw}{dt}$.Free
  4. Q4Let $U(x,y,z)=xyz,\ x=e^{-t},\ y=e^{-t}\cos t,\ z=\sin t,\ t\in\mathbb R$. Find $\dfrac{dU}{dt}$.Preview
  5. Q5If $w(x,y)=6x^3-3xy+2y^2,\ x=e^s,\ y=\cos s,\ s\in\mathbb R$, find $\dfrac{dw}{ds}$, and evaluate at $s=0$.Preview
  6. Q6If $z(x,y)=x\tan^{-1}(xy),\ x=t^2,\ y=se^t,\ s,t\in\mathbb R$. Find $\dfrac{\partial z}{\partial s}$ and $\dfrac{\partial z}{\partial t}$ at…Preview
  7. Q7Let $U(x,y)=e^x\sin y$, where $x=st^2,\ y=s^2t,\ s,t\in\mathbb R$. Find $\dfrac{\partial U}{\partial s},\dfrac{\partial U}{\partial t}$ and…Preview
  8. Q8Let $z(x,y)=x^3-3x^2y^3$, where $x=se^t,\ y=se^{-t},\ s,t\in\mathbb R$. Find $\dfrac{\partial z}{\partial s}$ and $\dfrac{\partial z}{\parti…Preview
  9. Q9$W(x,y,z)=xy+yz+zx,\ x=u-v,\ y=uv,\ z=u+v,\ u,v\in\mathbb R$. Find $\dfrac{\partial W}{\partial u},\dfrac{\partial W}{\partial v}$, and eval…Preview
8.6.2

Homogeneous Functions and Euler's Theorem

Definition 8.12 (Homogeneous Function).

+Exercise 8.7i6 questions
  1. Q1In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree. (i) $f(x,y)=x^…Free
  2. Q2Prove that $f(x,y)=x^3-2x^2y+3xy^2+y^3$ is homogeneous; what is the degree? Verify Euler's Theorem for $f$.Free
  3. Q3Prove that $g(x,y)=x\log\left(\dfrac{y}{x}\right)$ is homogeneous; what is the degree? Verify Euler's Theorem for $g$.Preview
  4. Q4If $u(x,y)=\dfrac{x^2+y^2}{\sqrt{x+y}}$, prove that $x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y}=\dfrac32 u$.Preview
  5. Q5If $v(x,y)=\log\left(\dfrac{x^2+y^2}{x+y}\right)$, prove that $x\dfrac{\partial v}{\partial x}+y\dfrac{\partial v}{\partial y}=1$.Preview
  6. Q6If $w(x,y,z)=\log\left(\dfrac{5x^3y^4+7y^2xz^4-75y^3z^4}{x^2+y^2}\right)$, find $x\dfrac{\partial w}{\partial x}+y\dfrac{\partial w}{\partia…Preview
8.7

Objective Type Questions

This 15-question multiple-choice self-test draws on every idea developed in the chapter — percentage error propagation, differentials of a single-variable function, partial derivatives (including seco…

+Exercise 8.8i15 questions
  1. Q1A circular template has a radius of $10$ cm. The measurement of radius has an approximate error of $0.02$ cm. Then the percentage error in c…Free
  2. Q2The percentage error of fifth root of $31$ is approximately how many times the percentage error in $31$? (1) $\dfrac1{31}$ (2) $\dfrac15$ (3…Free
  3. Q3If $u(x,y)=e^{x^2+y^2}$, then $\dfrac{\partial u}{\partial x}$ is equal to (1) $e^{x^2+y^2}$ (2) $2xu$ (3) $x^2u$ (4) $y^2u$Free
  4. Q4If $v(x,y)=\log(e^x+e^y)$, then $\dfrac{\partial v}{\partial x}+\dfrac{\partial v}{\partial y}$ is equal to (1) $e^x+e^y$ (2) $\dfrac{1}{e^x…Preview
  5. Q5If $w(x,y)=x^y,\ x>0$, then $\dfrac{\partial w}{\partial x}$ is equal to (1) $x^y\log x$ (2) $y\log x$ (3) $yx^{y-1}$ (4) $x\log y$Preview
  6. Q6If $f(x,y)=e^{xy}$, then $\dfrac{\partial^2 f}{\partial x\,\partial y}$ is equal to (1) $xye^{xy}$ (2) $(1+xy)e^{xy}$ (3) $(1+y)e^{xy}$ (4)…Preview
  7. Q7If we measure the side of a cube to be $4$ cm with an error of $0.1$ cm, then the error in our calculation of the volume is (1) $0.4$ cu.cm…Preview
  8. Q8The change in the surface area $S=6x^2$ of a cube when the edge length varies from $x_0$ to $x_0+dx$ is (1) $12x_0+dx$ (2) $12x_0dx$ (3) $6x…Preview
  9. Q9The approximate change in the volume $V$ of a cube of side $x$ metres caused by increasing the side by $1\%$ is (1) $0.3xdx\ m^3$ (2) $0.03x…Preview
  10. Q10If $g(x,y)=3x^2-5y+2y^2,\ x(t)=e^t$ and $y(t)=\cos t$, then $\dfrac{dg}{dt}$ is equal to (1) $6e^{2t}+5\sin t-4\cos t\sin t$ (2) $6e^{2t}-5\…Preview
  11. Q11If $f(x)=\dfrac{x}{x+1}$, then its differential is given by (1) $\dfrac{-1}{(x+1)^2}dx$ (2) $\dfrac{1}{(x+1)^2}dx$ (3) $\dfrac{1}{x+1}dx$ (4…Preview
  12. Q12If $u(x,y)=x^2+3xy+y-2019$, then $\dfrac{\partial u}{\partial x}\Big|_{(4,-5)}$ is equal to (1) $-4$ (2) $-3$ (3) $-7$ (4) $13$Preview
  13. Q13Linear approximation for $g(x)=\cos x$ at $x=\dfrac{\pi}{2}$ is (1) $x+\dfrac{\pi}{2}$ (2) $-x+\dfrac{\pi}{2}$ (3) $x-\dfrac{\pi}{2}$ (4) $-…Preview
  14. Q14If $w(x,y,z)=x^2(y-z)+y^2(z-x)+z^2(x-y)$, then $\dfrac{\partial w}{\partial x}+\dfrac{\partial w}{\partial y}+\dfrac{\partial w}{\partial z}…Preview
  15. Q15If $f(x,y,z)=xy+yz+zx$, then $f_x-f_z$ is equal to (1) $z-x$ (2) $y-z$ (3) $x-z$ (4) $y-x$Preview
8.8

Summary

- Let be differentiable and . The linear approximation of at is for all . - Absolute error Actual value Approximate value. . .

Sample & Board Papers

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

+Show 30 questions30 questions
  1. Q1If $V=ze^{ax+by}$ and $z$ is a homogeneous function of degree $n$ in $x$ and $y$, prove that $x\dfrac{\partial V}{\partial x} + y\dfrac{\par…Preview
  2. Q2If $u=\sin 3x\cos 4y$ then, verify $\dfrac{\partial^2 u}{\partial x\partial y} = \dfrac{\partial^2 u}{\partial y\partial x}$.Preview
  3. Q3If $u = f(x, y)$ is a differentiable function of $x$ and $y$; where $x$ and $y$ are differentiable functions of 't' then : (a) $\dfrac{du}{d…Preview
  4. Q4The radius of a circular disc is given as 24 cm. with a maximum error in measurement of 0.02 cm. Estimate the maximum error in the calculate…Preview
  5. Q5If $u = \dfrac{x}{y^2} - \dfrac{y}{x^2}$ then verify that $\dfrac{\partial^2 u}{\partial x \partial y} = \dfrac{\partial^2 u}{\partial y \pa…Preview
  6. Q6The differential of $y$ if $y = \sqrt{x^4 + x^2 + 1}$ is : (a) $\dfrac{1}{2}(4x^3 + 2x)^{-\frac{1}{2}}$ (b) $\dfrac{1}{2}(4x^3 + 2x)^{-\frac…Preview
  7. Q7If $u = \sin^{-1}\left(\dfrac{\sqrt{x} + \sqrt{y}}{\sqrt{x} - \sqrt{y}}\right) \cdot \tan\left(\dfrac{x^3 + y^3}{x^3 - y^3}\right)$ then fin…Preview
  8. Q8The percentage error in the $11^{th}$ root of the number 28 is approximately ________ times the percentage error in 28. (a) $11$ (b) $28$ (c…Preview
  9. Q9If $f(x,y) = \dfrac{1}{\sqrt{x^2+y^2}}$ then, prove that $x\dfrac{\partial f}{\partial x} + y\dfrac{\partial f}{\partial y} = -f$.Preview
  10. Q10(a) If $w = x+2y+z^2$ and $x=\cos t$; $y=\sin t$; $z=t$ find $\dfrac{dw}{dt}$ by using chain rule. Also find $\dfrac{dw}{dt}$ by substitutio…Preview
  11. Q11If $u(x, y) = e^{x^2+y^2}$, then $\dfrac{\partial u}{\partial x}$ is equal to : (a) $y^2u$ (b) $e^{x^2+y^2}$ (c) $2xu$ (d) $x^2u$Preview
  12. Q12For the function $f(x)=x^2+3x$, calculate the differential $df$ when $x=2$ and $dx=0.1$.Preview
  13. Q13If $U=\log(x^3+y^3+z^3)$ then find $\dfrac{\partial U}{\partial x}+\dfrac{\partial U}{\partial y}+\dfrac{\partial U}{\partial z}$.Preview
  14. Q14If $f(x)=\dfrac{x}{x+1}$, then its differential is : (a) $\dfrac{1}{x+1}dx$ (b) $\dfrac{-1}{(x+1)^2}dx$ (c) $\dfrac{-1}{x+1}dx$ (d) $\dfrac{…Preview
  15. Q15Find $df$ for $f(x)=x^2+3x$ and evaluate it for $x=2$ and $dx=0.1$.Preview
  16. Q16Assume that the cross section of the artery of human is circular. A drug is given to a patient to dilate his arteries. If the radius of an a…Preview
  17. Q17If the radius of a sphere with radius 10 cm, has to decrease by 0.1 cm, approximately how much will its volume decrease ?Preview
  18. Q18Use the linear approximation to find an approximate value of $(123)^{2/3}$.Preview
  19. Q19If $f(x)=\dfrac{x}{x+1}$, then its differential is given by : (a) $\dfrac{1}{x+1}\,dx$ (b) $\dfrac{-1}{(x+1)^2}\,dx$ (c) $\dfrac{-1}{x+1}\,d…Preview
  20. Q20Find $df$ for $f(x)=x^2+3x$ and evaluate it for $x=3$ and $dx=0.02$.Preview
  21. Q21If $u(x, y)=\dfrac{x^2+y^2}{\sqrt{x+y}}$, prove that $x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y}=\dfrac32 u$.Preview
  22. Q22(a) A conical water tank with vertex down of 12 meters height has a radius of 5 meters at the top. If water flows into the tank at a rate 10…Preview
  23. Q23If $u(x, y)=e^{x^2+y^2}$, then $\dfrac{\partial u}{\partial x}$ is equal to : (a) $x^2u$ (b) $e^{x^2+y^2}$ (c) $y^2u$ (d) $2xu$Preview
  24. Q24If $f(x)>0$ for all $x$ and $g(x)=\log(f(x))$, then $dg$ is : (a) $\dfrac{1}{f(x)}\,dx$ (b) $\dfrac{1}{f(x)}f'(x)\,dx$ (c) $\dfrac1x\,dx$ (d…Preview
  25. Q25Show that $F(x, y)=\dfrac{x^2+5xy-10y^2}{3x+7y}$ is a homogeneous function of degree 1.Preview
  26. Q26If $u(x, y)=x^2y+3xy^4$, $x=e^t$ and $y=\sin t$, find $\dfrac{du}{dt}$Preview
  27. Q27The percentage error of fifth root of 31 is approximately how many times the percentage error in 31 ? (a) $5$ (b) $\dfrac{1}{31}$ (c) $31$ (…Preview
  28. Q28Let $A=\{(x, y)\mid a<x<b,\ c<y<d\}\subset R^2$. If the function $u:A\to R^2$ is harmonic in A, then : (a) $\dfrac{\partial^2u}{\partial x^2…Preview
  29. Q29If $f(x, y)=\cos^{-1}\left(\dfrac{x}{y}\right)$, then show that $f_y=\dfrac{x}{y\sqrt{y^2-x^2}}$.Preview
  30. Q30Use the linear approximation to find approximate value of $\sqrt[4]{15}$Preview