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Mathematics · 2026 · 2 marks
CBSE 2026 · Region 5 · Set 1 · Q22
(a) Differentiate x^x with respect to x log x.
(a)
Differentiate $\displaystyle \mathrm{x}^{\mathrm{x}}$ with respect to x log x.
OR
(b)
If $\displaystyle \mathrm{y}=\mathrm{P} \cos \mathrm{ux}+\mathrm{Q} \sin \mathrm{ux}$, show that $\displaystyle \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+\mathrm{u}^{2} \mathrm{y}=0$.
Marking-scheme solution
(a)
Let $\displaystyle u=x^{x}$ and $\displaystyle v=x \log x$
$\displaystyle \dfrac{du}{dx}=x^{x}(1+\log x)$
$\displaystyle \dfrac{dv}{dx}=1+\log x$
$\displaystyle \therefore \dfrac{du}{dv}=x^{x}$
OR
(b)
$\displaystyle \dfrac{dy}{dx}=-P u \sin u x+Q u \cos u x$
$\displaystyle \dfrac{d^{2} y}{dx^{2}}=-P u^{2} \cos u x-Q u^{2} \sin u x$
$\displaystyle \dfrac{d^{2} y}{dx^{2}}=-u^{2}(P \cos u x+Q \sin u x)$
$\displaystyle \therefore \dfrac{d^{2} y}{dx^{2}}+u^{2} y=0$
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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.