Differentiate the following from first principle: `3^(x^2)`.

Explanation

`f(x)=3^(x^2)`


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


`f'(x)=lim_(h->0)(3^((x+h)^2)-3^(x^2))/(h)`


`f'(x)=lim_(h->0)(3^(x^2+h^2+2xh)-3^(x^2))/(h)`


`f'(x)=3^(x^2)*lim_(h->0)(3^(h^2+2xh)-1)/(h)`


We know `lim_(x->0) (a^x-1)/(x)=loga`


`f'(x)=3^(x^2)log3*lim_(h->0) (h^2+2xh)/(h)`


`f'(x)=3^(x^2)log3*lim_(h->0) h+2x`


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