# Explore y=a^x including calculus

- Author:
- Richard Trimble

- Topic:
- Calculus

**Investigate the shape of the graph**

**by varying the constant**

**below.**This curve is typical of many types of exponential growth or decay in nature such as early stage bacterial growth or radioactive decay. It also comes up in financial contexts with, say, a specific rate of interest applied to an account. If we wish to differentiate this function we need to return to first principles.

**Choose different values of**

**and use for example**

**to estimate this constant. What do you notice?**There are two other geometric ways of understanding this limit. - Firstly if we divide the value of the gradient at a point by the value of the function at that point the

**Use the interactivity below to explore the value of this limit for different values of**

**using either or both of these geometric understandings.**

**Is there a value of a for which the limit is equal to one?**Stop and think for a moment what that would mean, if there was such a value then it would mean that the derivative of that exponential function was equal to itself! To determine such a value we can cheat slightly and require the limit to equal one for a particular value of

**Try small values of h in this expression. What sort of answers do you get for this special value of**

**?**In fact we can find an explicit expression for this value just by using the binomial expansion. Remember that if

**Try this expression with some large values of**

**and see what you get.**

If you are familiar with the binomial expansion we can imagine what we would get if we binomially expanded this expression.
Some of the s cancel and we are left with expressions like .
as large as we please and overwhelm the deficit on the top line.
What does this mean for our expansion? If all the expressions like then all we are left with is the coefficients.
i.e. is the sum of the reciprocals of all the factorials.

**What happens to an expression like****or****as****?****Does the same thing happen for****?**It turns out that any expression like this does tend to one because we can always make