Differentiate each of the following from first principle: `a^(sqrt(x))`

Explanation

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

`frac{d}{d x}(a^{sqrt{x}})=lim _{h -> 0} frac{a^{sqrt{x+h}}-a^{sqrt{x}}}{h}`

`=lim _{h -> 0} frac{a^{sqrt{x}}(a^{sqrt{x+h}}-sqrt{x}-1)}{(x+h)-(x)}`

`=a^{sqrt{x}} lim _{h -> 0} frac{(a^{sqrt{x+h}}-sqrt{x}-1)}{(sqrt{x+h})^{2}-(sqrt{x})^{2}}`

`=a^{sqrt{x}} lim _{h -> 0} frac{(a^{sqrt{x+h}}-sqrt{x}-1)}{(sqrt{x+h}-sqrt{x})(sqrt{x+h}+sqrt{x})}`

`=a^{sqrt{x}} lim _{h -> 0} frac{(a^{sqrt{x+h}}-sqrt{x}-1)}{(sqrt{x+h}-sqrt{x})} lim _{h -> 0} frac{1}{(sqrt{x+h}+sqrt{x})}`

`=a^{sqrt{x}} log _{e} a frac{1}{2 sqrt{x}}`

`=frac{1}{2 sqrt{x}} a^{sqrt{x}} log _{e} a`