Differentiate `logsinx` from the first principles.

Explanation

Given,f(x)=logsinx.

Then, `f(x+h)=logsin(x+h)`

⇒`d/dxf(x)=lim_(h->0)(f(x+h)−f(x))/h`

⇒`d/dxf(x)=lim_(h->0)(logsin(x+h)−logsin(x))/h`

=`lim_(h->0)(log(sin(x+h)/sinx)/h)`

=`lim_(h->0)log(1+sin(x+h)/sinx−1)/h`

=`lim_(h->0)log(1+(sin(x+h)−sinx)/sinx)/h`

=`limh->0log(1+(sin(x+h)−sinx)/sinx)/(h(sin(x+h)−sinx)/sinx)×[(sin(x+h)−sinx)/sinx]`

...