Question

How to find instantaneous rate of change for `f(x)=e^{x}` at x=1?

Answer 1

The instantaneous rate of change is also known as the derivative. It is analogous to the slope of the tangent line at a point, as well.

We might say that the instantaneous rate of change of `f` at `x=a` is equal to `f'(a)`.

Here, we have to know that the derivative of `e^x` is itself—it's a unique and very important function, especially in calculus. That is:

`f(x)=e^x" "=>" "f'(x)=e^x`

So, the instantaneous rate of change of `f` at `x=1` is `f'(1)`, and we see that:

`f'(x)=e^x" "=>" "f'(1)=e^1=e`

The instantaneous rate of change of `f` at `x=1` is `e`, which is a transcendental number approximately equal to `2.7182818`.