✓ Board-verified
Mathematics · 2026 · 3 marks
CBSE 2026 · Region 3 · Set 1 · Q30
If ( sin x )^y = y^cos x, then find dy/dx.
If $\displaystyle (\sin \mathrm{x})^{\mathrm{y}}=\mathrm{y}^{\cos \mathrm{x}}$, then find $\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}$.
Marking-scheme solution
$\displaystyle (\sin x)^{y}=y^{\cos x} \Rightarrow y \log \sin x=\cos x \log y$
On diff. wrt x both sides, we get
$\displaystyle \dfrac{y \cos x}{\sin x}+\log \sin x \dfrac{dy}{dx}=\dfrac{\cos x}{y} \dfrac{dy}{dx}-\log y \cdot \sin x$
$\displaystyle \Rightarrow \dfrac{dy}{dx}=\dfrac{y\left(y \cos x+\log y \cdot \sin^{2} x\right)}{\sin x(\cos x-y \log \sin x)}$ or $\displaystyle \dfrac{(y \cot x+\sin x \cdot \log y) y}{\cos x-y \log \sin x}$
All Continuity and Differentiability questions2026 question paperNCERT solutions: Continuity and DifferentiabilityPractice Continuity and Differentiability →
More from Continuity and Differentiability
- The derivative of sin (x^2) w.r.t. x, at x=√π is:2024
- The value of k for which function f(x)= {x^2, x ≥ 0; kx, x<0} is differentiable at x=0 is:2023
- If f(x)= {(sin^2 a x)/(x^2), x ≠ 0; 1, x=0} is continuous at x=0, then the value of a is:2025
- Differentiate y=√(log { sin ((x^3)/3-1)}) with respect to x.2025
- If y=(cos x- sin x)/(cos x+ sin x), then (d y)/(d x) is:2023
- If f(x)= {(log (1+a x)+ log (1-b x))/x, for x ≠ 0; k, for x=0} is continuous at x=0, then the value of k is:2025
- 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 x^2)=(cos x)/((1- sin x)^2).2023
- If f(x)= {(x^2-4 x-5)/(x+1), x ≠-1; k, x=-1} is continuous at x=-1, then the value of k is:2026
CBSE Class 12 Mathematics past-paper question from the 2026 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.