Differentiation y=a^x

y=a^x

To find the derivative of y=a^x, we use the exact same steps as that used for differentiating y=e^x, and y=x^x as well. Hence, if you did those earlier you should be able to do this one.

Taking log on both sides.

Just as before, you take the log on both sides.

This brings the x down from the power position, as shown on the RHS. This is a basic log rule if you remember your algebra.

Differentiating both sides.

We then differentiate both sides using d/dx.

Since ln y has to be differentiated with respect to y we use an ancillary trick by using the following technique.

dy/dx × d/dy = d/dx

As you can see, the dy cancels out and essentially we still have d/dx. On the RHS, we can place ln a behind d/dx because it is just a constant.

Obviously d/dy ln y = 1/y which is a standard proof that you can use.

dy/dx

We can now move y to the RHS to give the expression shown above.

Solution

Since we know what y is, we can substitute that in to give the final answer. Hence the derivative of a^x = a^xln a

Mr Smith showed this to me in the late 80s, but if you find anyone showing you a proof in this way, then they have to be a good teacher!

This Article Continues...

Differentiation
Differentiation Chain Rule
Differentiation Quotient Rule
Differentiation Product Rule
Differentiation Formulas
Differentiation of ln x
Differentiation of Exponential
Differentiation of tan x
Differentiation of log x
Differentiation x^x
Differentiation y=a^x
Differentiation from First Principles