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Limits and Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits and Derivatives5 Marks
Question
Differentiate the functions with respect to 'x'.
$\text{x}\cos\text{x}$
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Answer

Let $\text{y}=\text{x}\cos\text{x}\ ...(\text{i})$
$\text{y}+\Delta\text{y}=(\text{x}+\Delta\text{x})\cos(\text{x}+\Delta\text{x})\ ...(\text{ii})$
Subtracting eq. (i) from eq. (ii)
$\text{y}+\Delta\text{y}-\text{y}=(\text{x}+\Delta\text{x})\cos(\text{x}+\Delta\text{x})-\text{x}\cos\text{x}$ 
$\Delta\text{y}=\text{x}\cos(\text{x}+\Delta\text{x})+\Delta\text{x}\cos(\text{x}+\Delta\text{x})-\text{x}\cos\text{x}$
Dividing both sides by taken the limit, we get
$=\lim\limits_{\Delta\text{x} \rightarrow 0}\frac{\Delta\text{y}}{\Delta\text{x}}$
$=\lim\limits_{\Delta\text{x} \rightarrow 0}\frac{\text{x}\cos(\text{x}+\Delta\text{x})-\text{x}\cos\text{x}+\Delta\text{x}\cos(\text{x}+\Delta\text{x})}{\Delta\text{x}}$
$=\lim\limits_{\Delta\text{x} \rightarrow 0}\frac{\text{x}\big[\cos(\text{x}+\Delta\text{x})-\cos\text{x}\big]}{\Delta\text{x}}$ 
$=\lim\limits_{\Delta\text{x} \rightarrow 0}\frac{\text{x}\bigg[-2\sin\frac{(\text{x}+\Delta\text{x}+\text{x})}{2}\cdot\sin\frac{(\text{x}+\Delta\text{x}-\text{x})}{2}\bigg]}{\Delta\text{x}}$
$=\lim\limits_{\Delta\text{x} \rightarrow 0}\frac{\text{x}\bigg[-2\sin\Big(\text{x}+\frac{\Delta\text{x}}{2}\Big)\cdot\sin\frac{\Delta\text{x}}{2}\bigg]}{2\times\frac{\Delta\text{x}}{2}}$
$\frac{\Delta\text{x}}{2}\rightarrow0$ Taking the limits, we have
$=\text{x}[-\sin\text{x}]+\cos\text{x}$
$=-\text{x}\sin\text{x}+\cos\text{x}$
Hence, the required answer is $-\text{x}\sin\text{x}+\cos\text{x}.$

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