Question

What is the derivative of `7^x`?

Answer 1

`color(red)(d/(dx)(a^x)=a^x*lna`
`:.y=7^x=>y^'=7^x*ln7`

Explanation:

We know that
`color(blue)(f^'(x)=lim_(t to x)(f(t)-f(x))/(t-x))`,where `f(x)=7^x`
`=lim_(t to x)(7^t-7^x)/(t-x)`
`=7^xlim_(t to x)(7^(t-x)-1)/(t-x)`,
Take, `(t-x)=h`, then, `t to x=>h to0`
`f^'(x)=7^xlim_(h to 0)(7^h-1)/h`,and, `color(red)(lim_(h to 0)(a^h-1)/h=lna`
`f^'(x)=7^x*ln7`