Differentiate `logsinx` by first principles.

Explanation

`f(x)=log sin x`.

Then, `f(x+h)=log sin (x+h)`

`Rightarrow frac{d}{d x} f(x)=lim _{h Rightarrow 0} frac{f(x+h)-f(x)}{h}`

`Rightarrow frac{d}{d x} f(x)=lim _{h Rightarrow 0} frac{log sin (x+h)-log sin (x)}{h}`

`=lim _{h Rightarrow 0} frac{log (frac{sin (x+h)}{sin x})}{h}`

`=lim _{h Rightarrow 0} frac{log (1+frac{sin (x+h)}{sin x}-1)}{h}`

`=lim _{h Rightarrow 0} frac{log (1+frac{sin (x+h)-sin x}{sin x})}{h}`

`=lim _{h Rightarrow 0} frac{log (1+frac{sin (x+h)-sin x}{sin x})}{h(frac{sin (x+h)-sin x}{sin x})} times[frac{sin (x+h)-sin x}{sin x}]`

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