Differentiate the following function from first principles: `e^(a x+b)`

Explanation

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


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


`=lim _{h -> 0} frac{e^{a x+b} e^{a h}-e^{a x+b}}{h}`


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


`=a e^{a x+b} lim _{h -> 0} frac{e^{a h}-1}{a h}`


`=a e^{a x+b}(1)`


`=a e^{a x+b}`