Continuity and Differentiability — MATHS STD 12 Science — Question
Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity and Differentiability2 Marks
Question
Differentiate $x^{\sin x}, x > 0 \text{w.r.t. x}.$
✓
Answer
Let $y = x^{\sin x}.$
Taking logarithm on both sides, we have
$\log y = \sin x \times logx$
Therefore $\frac{1}{y} \cdot \frac{d y}{d x}=\sin x \frac{d}{d x}(\log x)+\log x \frac{d}{d x}(\sin x)$
or $\frac{1}{y} \frac{d y}{d x}=(\sin x) \frac{1}{x} + \log x \cos x$
or $\frac{d y}{d x}=y\left[\frac{\sin x}{x}+\cos x \log x\right]$
or $\frac{d y}{d x} = x^{\sin x}\left[\frac{\sin x}{x}+\cos x \log x\right]$
or $\frac{d y}{d x} = x ^{sinx-1} \sin x + x^{\sin x} . \cos x \log x$
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