Find the equation of an exponential function knowing that its gradient=y coordinate

Question

Find the equation of an exponential function knowing that its gradient=y coordinate.

$f(x)=k^x$ What values would have to take $k$?

The $y$ coordinate has to be equal to the gradient, e.g. in $y=1$, the gradient has to be $1$.

Answer 1

If you are given that

$$f(x)=k^x$$

then $$f'(x)=k^x\log k=f(x)$$ is achieved when $$\log k=1.$$