Differentiation of Exponential

Differentiating an exponential function such as y=e^x is an easy one to do as you will see. This is a basic textbook style proof that I seem to remember from the 80s. If you ever find a teacher who shows you the proof like this, then that is a sign of an excellent teacher.

Whenever you have x as a power a nice trick to use is to take the log on both sides, and therefore on the RHS ln e^x becomes xln e, therefore bringing x down.

Then, simply differentiate both sides by d/dx thereby keeping the balance.

Since ln y differentiates with respect to y we can use a little ancillary trick using d/dy × dy/dx. As you can see, the dy cancels and the expression is still d/dx, however it provides the mechanism to differentiate ln y. On the RHS ln e =1, so the expression on the RHS is 1.

However, d/dy lny = 1/y, is a standard proof that can be used on the LHS.

We can now multiply both sides by y to give us dy/dx=y

Amazingly, we get the same result that we started with.

This Article Continues...