If $y = \sqrt{\sin x + y}$, then find $\frac{dy}{dx}$.
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Continuity, Differentiability and Differentiation - ISC Class 12 Mathematics Questions with Answers, Page 3
72 past-paper questions on Continuity, Differentiability and Differentiation from ISC Class 12 Mathematics papers (2027-2017), newest first, in full. Questions 41-60 are on this page, 20 to a page. Tap "Show answer" under a question to see its answer.
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If $\sin^{-1} x + \sin^{-1} y = \frac{\pi}{2}$, then $\frac{dy}{dx}$ is equal to
- (a)$\frac{x}{y}$
- (b)$-\frac{x}{y}$
- (c)$\frac{y}{x}$
- (d)$-\frac{y}{x}$
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The second derivative of $y = x^3 - 5x^2 + x$ is:
- (a)$10x - 5$
- (b)$6x - 10$
- (c)$3x^2 - 10x$
- (d)$3x^2 - 10x + 1$
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The value of $\lim_{x \to 0} \frac{\log(1+x)}{x}$ is equal to:
- (a)e
- (b)0
- (c)1
- (d)-1
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If $\sin^{-1} x + \sin^{-1} y = \frac{\pi}{2}$, then $\frac{dy}{dx}$ is equal to:
- (a)$\frac{x}{y}$
- (b)$-\frac{x}{y}$
- (c)$\frac{y}{x}$
- (d)$-\frac{y}{x}$
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If $y = t^2$ and $t = x + 3$ then $\frac{dy}{dx}$ is equal to:
- (a)$(x + 3)^2$
- (b)$2(x + 3)$
- (c)$2t$
- (d)$2(x + 3)^2$
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What will be the derivative of $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ with respect to $\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$?
- (a)-1
- (b)1
- (c)2
- (d)4
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The set of points, where the function $f(x) = x |x|$ is differentiable in:
- (a)$(-\infty, \infty)$
- (b)$(-\infty, 0) \cup (0, \infty)$
- (c)$(0, \infty)$
- (d)$[0, \infty)$
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If $x = \tan\left(\frac{1}{a} \log y\right)$, prove that $(1 + x^2)\frac{d^2y}{dx^2} + (2x - a)\frac{dy}{dx} = 0$
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Differentiate the function with reference to $x$:
$f(x) = \cos^{-1}\left(\sqrt{\frac{1 - \cos x}{2}}\right)$
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Using L’Hospital’s Rule, evaluate: $\lim_{x \to 0} \frac{8^x - 4^x}{4x}$
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Find $\frac{dy}{dx}$, if $x = at^2$ and $y = 2at$.
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Find $\frac{dy}{dx}$ if $x^3 + y^3 = 3axy$.
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Using L’Hospital’s rule, evaluate: $\lim_{x \to 0} \frac{xe^x - \log(1 + x)}{x^2}$.
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If $y = e^{m\sin^{-1} x}$, prove that $(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = m^2 y$.
Given: $y = e^{m\sin^{-1} x}$
To show: $(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = m^2 y$
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Verify Rolle’s theorem for the function, $f(x) = -1 + \cos x$ in the interval $[0, 2\pi]$.
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Using L'Hospital's Rule, evaluate: $\lim_{x \to 0} \frac{8^x - 4^x}{4x}$
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Show that the function $f(x) = |x - 4|$, $x \in \mathbb{R}$ is continuous, but not differentiable at $x = 4$.
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$f(x) = \frac{x^2 - 9}{x - 3}$ is not defined at $x = 3$. What value should be assigned to $f(3)$ for continuity of $f(x)$ at $x = 3$?
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Verify the Lagrange's mean value theorem for the function:
$f(x) = x + \frac{1}{x}$ in the interval $[1, 3]$
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