Question

4. Find the derivative of sin x from first principle.

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Answer 1

Explanation:

`\text{By First Principle,}`

`\begin{align}(d)/(dx)\sin x&=\lim_(h\to0)(\sin(x+h)-\sin x)/(h)\\&=\lim_(h\to0)(\sin x\cos h+\sin h\cos x-\sin x)/(h)\\&=\lim_(h\to0)\left((\cos h-1)/(h)\cdot\sin x\right)+\lim_(h\to0)\left((\sin h)/(h)\cdot\cos x\right)\\&=\sin x\cdot\left(\lim_(h\to0)(\cos h-1)/(h)\right)+\cos x\cdot\lim_(h\to0)\left((\sin h)/(h)\right)\\&=\sin x\cdot 0+\cos x\cdot1\\&=\cos x\end{align}`