Differentiate the following functions with respect to `x` from first principles: `sqrt(a x+b)`

Explanation

Let f(x)=`sqrt(a x+b)`

then,f(x+h)=`sqrt(a(x+h)+b)`

we know that` f'(x)=lim_(h->0)((f(x+h)-f(x))/h)`

so,`f'(x)=lim_(h->0)(sqrt(a(x+h)+b)-sqrt(a x+b))/h`

=`lim_(h->0)(sqrt(a(x+h)+b)-sqrt(a x+b))/hxx((sqrt(a(x+h)+b)+sqrt(a x+b))/(sqrt(a(x+h)+b)+sqrt(a x+b)))`

=`lim_(h->0)(ax+ah+b-ax-b)/(h.(sqrt(ax+ah+b)-sqrt(a x+b)))`

=`lim_(h->0)a/(sqrt(ax+ah+b)-sqrt(a x+b))`

=`a/(sqrt(a x+b)+sqrt(a x+b))`

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