Mathematics · Class 11 Science
Ch 18Differentiation — Class 11 Mathematics, concept-first.
Suppose a car travels from Mumbai to Pune. At every instant its position is being displaced from the starting point (Mumbai). The average speed over the whole journey is but the car's speed is rarely constant — it is different at different moments.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
The Derivative — Definition
The derivative grew out of two of the four classic 17th-century calculus problems: the tangent line problem and the velocity problem. Both reduce to the same limiting construction.
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.
Introduction
Suppose a car travels from Mumbai to Pune. At every instant its position is being displaced from the starting point (Mumbai).
Definition of Derivative and Differentiability
Definition. Let be a function defined on an open interval containing the point . If exists, then is said to be differentiable at , and this limit value is called the derivative of at , written .
Derivative by Method of First Principle
Finding a derivative straight from its defining limit — without invoking any differentiation rule established later — is called finding the derivative by method of first principle (also called 'ab ini…
Derivatives of Some Standard Functions
This section proves six standard derivative formulas directly from the first-principles limit of the previous section.
Relationship Between Differentiability and Continuity
Theorem. Every differentiable function is continuous.
Theorem 1 — Derivative of Sum of Functions
Theorem 1. If and are differentiable functions of such that , then
Theorem 2 — Derivative of Difference of Functions
Theorem 2. If and are differentiable functions of such that , then The book leaves this proof as an exercise, since it is identical in structure to the sum-rule proof of Theorem 1: give an increment ,…
Theorem 3 — Derivative of Product of Functions
Theorem 3. If and are differentiable functions of such that , then
Theorem 4 — Derivative of Quotient of Functions
Theorem 4. If and are differentiable functions of such that where , then
Derivatives of Algebraic Functions
A short reference table of algebraic-function derivatives, for quick lookup: Each entry was already proved by first principles earlier in the chapter (the first two in 9.1.4, the third as a 'Try the f…
Derivatives of Trigonometric Functions
A reference table of all six trigonometric-function derivatives: Sine, tangent and secant were proved by first principles in section 9.1.4; cosine, cotangent and cosecant were proved as the 'Try the f…
Derivatives of Logarithmic and Exponential Functions
A reference table of logarithmic and exponential derivatives: All three were proved by first principles in section 9.1.4 ( and directly; as the 'Try the following' companion exercise, being the specia…
Brief Idea of L'Hospital's Rule
Brief idea of L'Hospital's Rule. Consider two functions and . If and (a indeterminate form), and if and where , then If instead too, then — provided as well — the same situation reappears one derivati…
More questions
75 Q+−Show 22 questionsHide questions22 questions
- Q1$x^2+3x-1$Free
- Q2$\sin(3x)$Free
- Q3$e^{2x+1}$Free
- Q4$3^x$Preview
- Q5$\log(2x+5)$Preview
- Q6$\tan(2x+3)$Preview
- Q7$\sec(5x-2)$Preview
- Q8$x\sqrt{x}$Preview
- Q9$\sqrt{2x+5}$ at $x=2$Preview
- Q10$\tan x$ at $x=\dfrac{\pi}{4}$Preview
- Q11$2^{3x+1}$ at $x=2$Preview
- Q12$\log(2x+1)$ at $x=2$Preview
- Q13$e^{3x-4}$ at $x=2$Preview
- Q14$\cos x$ at $x=\dfrac{5\pi}{4}$Preview
- Q15Show that the function $f$ is not differentiable at $x=-3$, where $f(x)=x^2+2$ for $x<-3$, $=2-3x$ for $x\ge -3$.Preview
- Q16Show that $f(x)=x^2$ is continuous and differentiable at $x=0$.Preview
- Q17Discuss the continuity and differentiability of $f(x)=x|x|$ at $x=0$.Preview
- Q18Discuss the continuity and differentiability of $f(x)=(2x+3)|2x+3|$ at $x=-3/2$.Preview
- Q19Discuss the continuity and differentiability of $f(x)=[x]$ at $x=2$, if $x\in[0,4)$. [where $[\,\cdot\,]$ is the greatest integer (floor) fu…Preview
- Q20Test the continuity and differentiability of $f(x)=3x+2$ if $x>2$, $=12-x^2$ if $x\le 2$, at $x=2$.Preview
- Q21If $f(x)=\sin x-\cos x$ if $x\le \pi/2$, $=2x-\pi+1$ if $x>\pi/2$. Test the continuity and differentiability of $f$ at $x=\pi/2$.Preview
- Q22Examine the function $f(x)=x^2\cos\!\left(\dfrac1x\right)$, for $x\ne 0$, $=0$ for $x=0$, for continuity and differentiability at $x=0$.Preview
+−Show 5 questionsHide questions5 questions
- Q23If $f(x)=\dfrac{1}{x^n}$, for $x\ne 0$, $n\in N$, then prove that $f'(x)=-\dfrac{n}{x^{n+1}}$.Free
- Q24If $f(x)=\cos x$, then prove that $f'(x)=-\sin x$.Free
- Q25If $f(x)=\cot x$, then prove that $f'(x)=-\text{cosec}^2x$.Preview
- Q26If $f(x)=\text{cosec}\,x$, then prove that $f'(x)=-\text{cosec}\,x\cdot\cot x$.Preview
- Q27If $f(x)=e^x$, then prove that $f'(x)=e^x$.Preview
+−Show 30 questionsHide questions30 questions
- Q28$y = x^{4/3} + e^x - \sin x$Free
- Q29$y = \sqrt{x} + \tan x - x^3$Free
- Q30$y = \log x - \text{cosec}\,x + 5^x - \dfrac{3}{x^{3/2}}$Free
- Q31$y = x^{7/3} + 5x^{4/5} - \dfrac{5}{x^{2/5}}$Preview
- Q32$y = 7^x + x^7 - \dfrac{2}{3}x\sqrt{x} - \log x + 7^7$Preview
- Q33$y = 3\cot x - 5e^x + 3\log x - \dfrac{4}{x^{3/4}}$Preview
- Q34$y = x^5\tan x$Preview
- Q35$y = x^3\log x$Preview
- Q36$y = (x^2+2)^2\sin x$Preview
- Q37$y = e^x\log x$Preview
- Q38$y = x^{3/2}\,e^x\log x$Preview
- Q39$y = \log\!\left(e^{x^3}\right)\cdot\log(x^3)$Preview
- Q40$y = x^2\sqrt{x} + x^4\log x$Preview
- Q41$y = e^x\sec x - x^{5/3}\log x$Preview
- Q42$y = x^4 + x\sqrt{x}\cos x - x^2 e^x$Preview
- Q43$y = (x^3-2)\tan x - x\cos x + 7^x\cdot x^7$Preview
- Q44$y = \sin x\log x + e^x\cos x - e^x\sqrt{x}$Preview
- Q45$y = e^x\tan x + \cos x\log x - \sqrt{x}\cdot 5^x$Preview
- Q46$y = \dfrac{x^2+3}{x^2-5}$Preview
- Q47$y = \dfrac{\sqrt{x}+5}{\sqrt{x}-5}$Preview
- Q48$y = \dfrac{xe^x}{x+e^x}$Preview
- Q49$y = \dfrac{x\log x}{x+\log x}$Preview
- Q50$y = \dfrac{x^2\sin x}{x+\cos x}$Preview
- Q51$y = \dfrac{5e^x-4}{3e^x-2}$Preview
- Q52If $f(x)$ is a quadratic polynomial such that $f(0)=3$, $f'(2)=2$ and $f'(3)=12$, then find $f(x)$.Preview
- Q53If $f(x)=a\sin x-b\cos x$, $f'\!\left(\dfrac{\pi}{4}\right)=\sqrt2$ and $f'\!\left(\dfrac{\pi}{6}\right)=2$, then find $f(x)$.Preview
- Q54Fill in the blanks (Activity Problem). $y=e^x.\tan x$, differentiate w.r.t. $x$: $\dfrac{dy}{dx}=\dfrac{d}{dx}(e^x\tan x)=\square\cdot\dfrac…Preview
- Q55Fill in the blanks (Activity Problem). $y=\dfrac{\sin x}{x^2+2}$, differentiate w.r.t. $x$: $\dfrac{dy}{dx}=\dfrac{\square\dfrac{d}{dx}(\sin…Preview
- Q56Fill in the blanks (Activity Problem). $y=(3x^2+5)\cos x$, differentiate w.r.t. $x$: $\dfrac{dy}{dx}=\dfrac{d}{dx}\big[(3x^2+5)\cos x\big]=(…Preview
- Q57Differentiate $\tan x$ and $\sec x$ w.r.t. $x$ using the formulae for differentiation of $\dfrac{u}{v}$ and $\dfrac1v$ respectively.Preview
+−Show 8 questionsHide questions8 questions
- Q58Select the appropriate option from the given alternative. If $y = \dfrac{x-4}{\sqrt{x}+2}$, then $\dfrac{dy}{dx} =$ (A) $\dfrac{1}{x+4}$ (B)…Free
- Q59If $y = \dfrac{ax+b}{cx+d}$, then $\dfrac{dy}{dx} =$ (A) $\dfrac{ab-cd}{(cx+d)^2}$ (B) $\dfrac{ax-c}{(cx+d)^2}$ (C) $\dfrac{ac-bd}{(cx+d)^2}…Free
- Q60If $y = \dfrac{3x+5}{4x+5}$, then $\dfrac{dy}{dx} =$ (A) $-\dfrac{15}{(3x+5)^2}$ (B) $-\dfrac{15}{(4x+5)^2}$ (C) $-\dfrac{5}{(4x+5)^2}$ (D)…Free
- Q61If $y = \dfrac{5\sin x-2}{4\sin x+3}$, then $\dfrac{dy}{dx} =$ (A) $\dfrac{7\cos x}{(4\sin x+3)^2}$ (B) $\dfrac{23\cos x}{(4\sin x+3)^2}$ (C…Preview
- Q62Suppose $f(x)$ is the derivative of $g(x)$ and $g(x)$ is the derivative of $h(x)$. If $h(x)=a\sin x+b\cos x+c$ then $f(x)+h(x)=$ (A) $0$ (B)…Preview
- Q63If $f(x)=2x+6$ for $0\le x\le 2$, $=ax^2+bx$ for $2<x\le 4$, is differentiable at $x=2$ then the values of $a$ and $b$ are. (A) $a=-\dfrac32…Preview
- Q64If $f(x)=x^2+\sin x+1$ for $x\le 0$, $=x^2-2x+1$ for $x\le 0$ then (A) $f$ is continuous at $x=0$, but not differentiable at $x=0$ (B) $f$ i…Preview
- Q65If $f(x)=\dfrac{x^{50}}{50}+\dfrac{x^{49}}{49}+\dfrac{x^{48}}{48}+\cdots+\dfrac{x^2}{2}+x+1$, then $f'(1)=$ (A) $48$ (B) $49$ (C) $50$ (D) $…Preview
+−Show 10 questionsHide questions10 questions
- Q66Determine whether the following function is differentiable at $x=3$ where, $f(x)=x^2+2$, for $x\ge 3$, $=6x-7$, for $x<3$.Free
- Q67Find the values of $p$ and $q$ that make function $f(x)$ differentiable everywhere on $R$: $f(x)=3-x$, for $x<1$, $=px^2+qx$, for $x\ge 1$.Free
- Q68Determine the values of $p$ and $q$ that make the function $f(x)$ differentiable on $R$ where $f(x)=px^3$, for $x<2$, $=x^2+q$, for $x\ge 2$…Free
- Q69Determine all real values of $p$ and $q$ that ensure the function $f(x)=px+q$ for $x\le 1$, $=\tan\!\left(\dfrac{\pi x}{4}\right)$, for $1<x…Preview
- Q70Discuss whether the function $f(x)=|x+1|+|x-1|$ is differentiable $\forall x\in R$.Preview
- Q71Test whether the function $f(x)=2x-3$, for $x\ge 2$, $=x-1$, for $x<2$, is differentiable at $x=2$.Preview
- Q72Test whether the function $f(x)=x^2+1$, for $x\ge 2$, $=2x+1$, for $x<2$, is differentiable at $x=2$.Preview
- Q73Test whether the function $f(x)=5x-3x^2$ for $x\ge 1$, $=3-x$, for $x<1$, is differentiable at $x=1$.Preview
- Q74If $f(2)=4$, $f'(2)=1$ then find $\displaystyle\lim_{x\to2}\left[\dfrac{xf(2)-2f(x)}{x-2}\right]$Preview
- Q75If $y = \dfrac{e^x}{\sqrt{x}}$ find $\dfrac{dy}{dx}$ when $x=1$.Preview