Continuity and Differentiability
NCERT Class 12
CBSE
Grade 12
Question:
Find $\dfrac{dy}{dx}$ if $y = (x \cos x)^x + (x \sin x)^{1/x}$.
Step-by-Step Solution
Set $y = u + v$. Differentiate $u = (x\cos x)^x \Rightarrow \dfrac{du}{dx} = (x\cos x)^x [ 1 - x\tan x + \log(x\cos x) ]$. [2.0 Marks]
Differentiate $v = (x\sin x)^{1/x} \Rightarrow \dfrac{dv}{dx} = (x\sin x)^{1/x} \left[ \dfrac{x\cot x + 1 - \log(x\sin x)}{x^2} \right]$. [2.0 Marks]
Total $\dfrac{dy}{dx} = \dfrac{du}{dx} + \dfrac{dv}{dx}$. [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Differentiating $u = (x\cos x)^x$: 2.0 Marks
Differentiating $v = (x\sin x)^{1/x}$: 2.0 Marks
Summing partial derivatives: 1.0 Mark
Correct Answer:
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