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The left-hand derivative of  \small f(x)=[x] \sin (\pi x) \text { at } x=k,  where  is an integer, is 

Option: 1

(-1)^{\mathrm{k}}(\mathrm{k}-1) \pi


Option: 2

(-1)^{\mathrm{k}-1}(\mathrm{k}-1) \pi


Option: 3

(-1)^{\mathrm{k}} \mathrm{k} \pi


Option: 4

(-1)^{\mathrm{k-1}} \mathrm{k} \pi


Answers (1)

best_answer

\begin{aligned} & f(x)=[x] \sin (\pi x) \\ & f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{(k-1) \sin \pi(k-h)}{-h} \\ & =\lim _{h \rightarrow 0} \frac{(k-1)(-1)^{k-1} \sin \pi h}{-h} \\ & =\pi(k-1)(-1)^k . \end{aligned}

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Rakesh

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