CBSE 2025 · Region 6 · Set 1 · Q27 · 3 marks
Differentiate $\displaystyle \mathrm{y}=\sin ^{-1}\left(3 \mathrm{x}-4 \mathrm{x}^{3}\right)$ w.r.t. x , if $\displaystyle \mathrm{x} \in\left[-\frac{1}{2}, \frac{1}{2}\right]$.Differentiate $\displaystyle \mathrm{y}=\cos ^{-1}\left(\frac{1-\mathrm{x}^{2}}{1+\mathrm{x}^{2}}\right)$ with respect to x , when $\displaystyle \mathrm{x} \in(0,1)$.
Differentiate $\displaystyle \mathrm{y}=\sin ^{-1}\left(3 \mathrm{x}-4 \mathrm{x}^{3}\right)$ w.r.t. x , if $\displaystyle \mathrm{x} \in\left[-\frac{1}{2}, \frac{1}{2}\right]$.
Differentiate $\displaystyle \mathrm{y}=\cos ^{-1}\left(\frac{1-\mathrm{x}^{2}}{1+\mathrm{x}^{2}}\right)$ with respect to x , when $\displaystyle \mathrm{x} \in(0,1)$.
Marking-scheme solution
$\displaystyle \mathrm{x}=\operatorname{sint}$ gives $\displaystyle \mathrm{y}=\sin ^{-1}(\sin 3 t)=3 t=3 \sin ^{-1} \mathrm{x}$
\[\frac{d \mathrm{y}}{d \mathrm{x}}=\frac{3}{\sqrt{1-\mathrm{x}^{2}}}
\]
Aliter: $\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{3-12 \mathrm{x}^{2}}{\sqrt{1-\left(3 \mathrm{x}-4 \mathrm{x}^{3}\right)^{2}}}$
$\displaystyle \mathrm{x}=\operatorname{tant}$ gives $\displaystyle \mathrm{y}=\cos ^{-1}(\cos 2 t)=2 t=2 \tan ^{-1} \mathrm{x}$
\[\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2}{1+\mathrm{x}^{2}}
\]
Aliter: $\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-1}{\sqrt{1-\left(\dfrac{1-\mathrm{x}^{2}}{1+\mathrm{x}^{2}}\right)^{2}}} \cdot \frac{-4 \mathrm{x}}{\left(1+\mathrm{x}^{2}\right)^{2}}$.
Continuity and DifferentiabilityDifferentiabilityApplyshort_answermedium
More from Continuity and Differentiability
- If f(x)=. is continuous at x=0, then the value of a is:2025 · asked 4×
- The value of k for which function f(x)=. is differentiable at x=0 is:2023 · asked 3×
- If y=( cos x- sin x)/( cos x+ sin x), then (d y)/(d x) is:2023 · asked 3×
- If f(x)=. is continuous at x=0, then the value of k is:2025 · asked 3×
- Differentiate sec^-1( 1√1-x^2) w.r.t. sin^-1(2 x √1-x^2). OR If y= tan x+ sec x, then prove that (d^2 y)/(d…2023 · asked 3×
- If f(x)=. is continuous at x=-1, then the value of k is:2026 · asked 3×
- The value of k for which the function f(x)= casesx^2 sin 1/x, & x ≠ 0 k(x+1), & x=0 cases is a continuous…2026 · asked 3×
- If tan ((x+y)/(x-y))=k, then (dy)/(dx) is equal to2023 · asked 3×
CBSE Class 12 Mathematics past-paper question from the 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.