Is d/dx (a^x)=x a^{x-1}? (a conceptual error in O/A Level Math)

In O Level, students are taught that \boxed{\frac{d}{dx}(x^{n})=nx^{n-1}}

So naturally, students may think that \displaystyle\frac{d}{dx}(a^{x})=xa^{x-1}?? (a is a constant)

Well, actually that is good pattern spotting, but unfortunately it is incorrect. Do not be too disheartened if you make this mistake, it is a common mistake.

The above is a conceptual error as  \boxed{\frac{d}{dx}(x^{n})=nx^{n-1}} only holds when n is a constant.

Fortunately, this question is rarely tested, though it is quite possible that it can come up in A Levels.

To fully understand the following steps, it would help read my other post (Why is e^(ln x)=x?) first.

First, we write \displaystyle a^x=e^{\ln a^x}=e^{x\ln a}.

\displaystyle    \begin{array}{rcl}    \frac{d}{dx} (a^x)&=&\frac{d}{dx} (e^{x\ln a})\\    &=&e^{x\ln a}(\ln a)\\    &=&e^{\ln a^x}(\ln a)\\    &=&a^x(\ln a)    \end{array}

After fully understanding the above steps, you may memorize the formula if you wish:

\boxed{\frac{d}{dx} (a^x)=a^x(\ln a)}

Memory Tip: If you let a=e, you should get \boxed{\frac{d}{dx} (e^x)=e^x(\ln e)=e^x}

The above steps involve the chain rule, which I will cover in a subsequent post.


Author: mathtuition88

Math and Education Blog

One thought on “Is d/dx (a^x)=x a^{x-1}? (a conceptual error in O/A Level Math)”

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

%d bloggers like this: