Tutoring calculus, you might be asked about the derivative of a^x from time to time. The math tutor shows the trick to this old favourite.

Example: Find the derivative of a^x

Solution:

As in so many cases, the trick here is to rewrite the expression to a form more easily penetrable by derivative techniques. With this example, we’ll realize that e^x and lnx are inverses. Therefore,

a^x=e^(lna^x)

Using the “bring the exponent down in front of a log” rule, we can proceed to

e^(lna^x)=e^(xlna)

Now, we recall the derivate rule

d/dx e^(f(x))=f'(x)e^(f(x))

Since lna is a constant, the derivate of xlna is just lna. Therefore,

d/dx e^(xlna)=(lna)e^(xlna)

Therefore,

d/dx a^x=(lna)e^(xlna)=(lna)a^x

I’ll be talking more about derivatives in future posts. Cheers:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.