Mathematics · Class 12 Science
Ch 5Continuity and Differentiability — Class 12 Mathematics, concept-first.
A function is intuitively continuous at a point if its graph can be traced through that point without lifting the pen -- there is no jump, no break, and no hole. Making this precise needs exactly the machinery of limits: three separate conditions must all hold together.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Second Order Derivatives
The second order derivative measures how the slope of is itself changing (physically, acceleration when is position). For an explicit function, differentiate twice in succession.
Most relevant Q&A
- If $y=e^x\sin x$, find $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}$, and show that $\dfrac{d^2y}{dx^2}-2\dfrac{dy}{dx}+2y=0$.Free
- If $y=x^3-3x^2+5x$, find $\dfrac{d^2y}{dx^2}$.Preview
- If $y=e^{2x}$, find $\dfrac{d^2y}{dx^2}$.Free
- If $y=\sin x$, find $\dfrac{d^2y}{dx^2}$ and verify that $\dfrac{d^2y}{dx^2}+y=0$.Preview
- If $x^2+y^2=r^2$ ($r$ constant), find $\dfrac{d^2y}{dx^2}$.Preview
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Continuity of a Function
A function is intuitively continuous at a point if its graph can be traced through that point without lifting the pen -- there is no jump, no break, and no hole.
Differentiability
Definition (differentiability at a point). A function is said to be differentiable at if the limit exists (finite).
Chain Rule
Many functions that arise in practice are not simple polynomials or single trigonometric expressions, but composite functions -- a function applied to the output of another function, such as or .
Derivatives of Inverse Trigonometric Functions
Each inverse trigonometric function is the inverse of a trigonometric function restricted to an interval on which that function is one-one, and its derivative can be found by writing the inverse relat…
Implicit Differentiation
So far every function differentiated has been given explicitly, in the form , with isolated on one side.
Exponential and Logarithmic Functions
Before their derivatives can be found (Section 7), the exponential and logarithmic functions themselves need a precise definition and a summary of their key properties.
Derivatives of Exponential and Logarithmic Functions
Derivative of . By the first-principles definition, and using the standard limit (established in the limits chapter), So is its own derivative -- the defining property of the natural exponential, and…
Logarithmic Differentiation
The problem this technique solves. None of the differentiation rules developed so far directly handle a function of the form , where both the base and the exponent are themselves functions of -- for i…
Parametric Differentiation
Parametric form. Sometimes and are both expressed in terms of a third, independent variable (the parameter), rather than one being written directly as a function of the other: As varies, the point tra…
Second Order Derivatives
Definition. If is differentiable, its derivative is itself a function of ; if that function is in turn differentiable, its derivative is called the second order derivative (or second derivative) of wi…
Summary
Continuity at : defined, exists, and -- equivalently .
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 30 questionsHide questions30 questions
- Q1If y = sin^-1 (2x / (1 + x^2)), find dy/dx.Preview
- Q2If f(2) = 4, f'(2) = 4, then evaluate lim (x -> 2) [x f(2) - 2 f(x)] / (x - 2).Preview
- Q3If cos y = x cos(a + y), (a not equal to 0), then show that dy/dx = cos^2(a + y) / sin a. **OR** If x = sin t, y = sin kt (k not equal to 0,…Preview
- Q4If y = tan^-1 [(5 - x) / (1 + 5x)], then value of dy/dx is (a) -1/(1+x^2) (b) 1/(1+x^2) (c) 5 (d) 5/(1+x^2)Preview
- Q5If f(x) = x for x ≥ 0, = 2 for x < 0, show that f(x) is discontinuous at x = 0.Preview
- Q6If ye^y = x, then show that dy/dx = y / (x(1 + y)).Preview
- Q7f(x) = |x|/x for x ≠ 0, = 0 for x = 0. Find the point of discontinuity of f(x). **OR** If y = sin(2 sin⁻¹ x), then show that (1 - x²) d²y/dx…Preview
- Q8If f(x) = log(3x + 1), then the value of f''(1) is: **OR** If f(x) = sin x / x (x ≠ 0) is continuous at x = 0, then the value of f(0) will b…Preview
- Q9Value of d/dx(cot x°) is equal to (a) -cosec²x° (b) cosec²x° (c) -(180/π)cosec²x° (d) -(π/180)cosec²x°Preview
- Q10Show that f(x) = |x| + 2 is continuous at x = 0.Preview
- Q11If y = a cos x - b sin x, then show that d²y/dx² + y = 0.Preview
- Q12If y = 1 + a/(x-a) + bx/((x-a)(x-b)) + cx²/((x-a)(x-b)(x-c)), then show that dy/dx = (y/x)[a/(a-x) + b/(b-x) + c/(c-x)]. **OR** If (x-a)² +…Preview
- Q13If dy/dx = y/x, then value of d²y/dx² is (a) 0 (b) y²/x² (c) y/x² (d) 1/xPreview
- Q14If f'(a) exists finitely, then show that f(x) is continuous at x = a.Preview
- Q15If x = t, y = t², then find d²y/dx².Preview
- Q16Define Rolle's theorem.Preview
- Q17If f(x) = 2 - x for x ≤ 0, = 2 + 2x for x > 0, show that f(x) is continuous at x = 0 but f'(0) does not exist. **OR** If y = log(tan(x/2)),…Preview
- Q18If G(x) = -√(25-x²), then Lt(x→1) [G(x)-G(1)]/(x-1) has the value (a) 1/24 (b) 1/5 (c) -√24 (d) 1/√24Preview
- Q19The function f(x) = x²/|x| for x≠0, and f(x) = 0 for x=0. Examine the continuity of the function f(x) at x=0.Preview
- Q20If y = tan⁻¹(secx+tanx), then find the value of d²y/dx² at x = π/4.Preview
- Q21Examine whether Rolle's theorem is applicable to f(x)=cotx in [-π/2, π/2].Preview
- Q22If f(x) = 3ax+b for x>1, 11 for x=1, 5ax-2b for x<1, is continuous at x=1, then find the values of a and b. **OR** If 2x = y^(1/m) + y^(-1/m…Preview
- Q23Let $f(x) = \dfrac{x^2 - 1}{x^3 - 1}$, when $x \neq 1$, is continuous at $x = 1$. Then the value of $f(1)$ is (a) $1$ (b) $\dfrac{1}{3}$ (c)…Preview
- Q24If $x^m y^n = (x + y)^{m+n}$, then the value of $\dfrac{dy}{dx}$ is (a) $0$ (b) $\dfrac{y}{x}$ (c) $\dfrac{x + y}{xy}$ (d) $xy$Preview
- Q25If $f(2) = 4$, $f'(2) = 4$, then the value of $\displaystyle\lim_{x \to 2} \dfrac{x f(2) - 2 f(x)}{x - 2}$ is (a) $-2$ (b) $2$ (c) $3$ (d) $…Preview
- Q26If $f(x) = \log_x (\log_e x)$, then the value of $f'(e)$ is (a) $e$ (b) $\dfrac{2}{e}$ (c) $\dfrac{1}{e}$ (d) $0$Preview
- Q27If $x = \sin^{-1} t$, $y = \sqrt{1 - t^2}$, then the value of $\dfrac{d^2 y}{dx^2}$ at $t = 1$ is (a) $1$ (b) $0$ (c) $\dfrac{1}{2}$ (d) $-1…Preview
- Q28If $\dfrac{dx}{dy} = l$ and $\dfrac{d^2 x}{dy^2} = m$, then the value of $\dfrac{d^2 y}{dx^2}$ is (a) $-\dfrac{m}{l^3}$ (b) $\dfrac{m}{l^3}$…Preview
- Q29Let $f(x) = \begin{cases} x^2 + ax + b, & x < 1 \\ x, & x \ge 1 \end{cases}$. If $f(x)$ is differentiable at $x = 1$, then $(a - b)$ is equa…Preview
- Q30The points of discontinuity of the function $f(x) = \dfrac{x^2 + 4x + 3}{x^3 + 3x^2 - x - 3}$ are (a) $x = 1, -1, -3$ (b) $x = -1, -3$ (c) $…Preview
More questions
36 Q+−Show 3 questionsHide questions3 questions
- Q34If $y=e^x\sin x$, find $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}$, and show that $\dfrac{d^2y}{dx^2}-2\dfrac{dy}{dx}+2y=0$.Free
- Q35If $x=at^2,\ y=2at$ (parametric form of a parabola), find $\dfrac{d^2y}{dx^2}$.Preview
- Q36If $y=x^x$ ($x>0$), find $\dfrac{dy}{dx}$ using logarithmic differentiation, and find the value of $x$ for which $\dfrac{dy}{dx}=0$.Preview
+−Show 9 questionsHide questions9 questions
- Example 1Discuss the continuity of $f(x)=\begin{cases}x^2+1, & x\le2\\ 3x-1, & x>2\end{cases}$ at $x=2$.Free
- Example 2Show that $f(x)=|x-1|$ is continuous at $x=1$ but not differentiable at $x=1$.Free
- Example 3Differentiate $y=\sin(3x^2+2x)$ with respect to $x$.Free
- Example 4Differentiate $y=\sin^{-1}(x^2)$ with respect to $x$.Preview
- Example 5If $x^2+y^2=25$, find $\dfrac{dy}{dx}$.Preview
- Example 6Differentiate $y=\ln(x^2+1)$ with respect to $x$.Preview
- Example 7If $y=x^x$ ($x>0$), find $\dfrac{dy}{dx}$.Preview
- Example 8If $x=at^2,\ y=2at$, find $\dfrac{dy}{dx}$.Preview
- Example 9If $y=x^3-3x^2+5x$, find $\dfrac{d^2y}{dx^2}$.Preview