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I'm just starting to learn about derivatives.

I've seen the derivative of x^2 given as 2x. (1)

But I've also seen the derivative of a^x given as a ^ x ln a. (2)

Suppose we use 3 as the base. From (1), we get that the derivative of 3^2 is 6.

But if I use (2) the a^x formula (with a = 3 and x = 2), I can't get a ^ x ln a to come out to 6.

I'm not sure if I'm supposed to be calculating a ^ (x ln a) or (a ^ x) ln a, but either way it doesn't come to 6.

What am I missing?

Many thanks!

Bruce
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    $x^2$ is a completely different function than $2^x$. If you use the limit definition of a derivative, you'll see that two derivations of the derivative play out pretty differently – gist076923 Nov 08 '22 at 13:15
  • The derivative of $a^x$ is $(\ln(a) ) a^x$. It is different from the derivative of $x^a$ which is $ax^{a-1}$. – Bumblebee Nov 08 '22 at 13:16
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    Also, if you have a function that you want to differentiate, for example $3^x$, you're not supposed to put anything for the value of $x$ before calculating the derivative. That's because if you plug in some value of $x$, then the outcome ($3^2$ in this case) is a constant, and the derivative of a constant is not the thing that you're after here. – Matti P. Nov 08 '22 at 13:17
  • For some basic information about writing mathematics at this site see, e.g., here, here, here and here. – Another User Nov 08 '22 at 13:23
  • Thanks for your comments. I'm starting to get it I think. – Bruce Nov 08 '22 at 13:32
  • "derivative of $3^2$ is $6$" - The derivative of any number (a constant function) is always $0$. The derivative of the squaring function $x\mapsto x^2$ is the doubling function $x\mapsto2x$. – mr_e_man Nov 08 '22 at 13:53
  • In case anyone stumbles across this with the same misunderstanding as me. You can have two functions that, for a particular value of x, give the same value of y - for example the function x^2 when x = 2, and the function a^x with a = 2 and x = 2 - yet the derivative is different. This is because the derivative is the slope of the tangent to the curve. Two slopes can pass through the same point - eg (2,4) - and yet have different slopes. The derivative is the slope of the tangent, and of course different functions (curves) have different tangents at the same point. – Bruce Nov 08 '22 at 14:41

1 Answers1

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The derivative of $f(x)^{g(x)}$ is $$f(x)^{g(x)}\left(g'(x)\ln f(x)+\frac{f'(x)g(x)}{f(x)}\right)$$

If $f(x)=x$ and $g(x)=2$, we get $$x^2\left(0+\frac2x\right)=2x$$

If $f(x)=a$ and $g(x)=x$, we get $$a^x(\ln a+0)=a^x\ln a$$

ajotatxe
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  • Thanks for your answer. But I'm a bit intrigued. If you look at my question, it's clear that I know nothing at all about derivatives - in fact I just started looking at them. How likely do you think I am to be able to understand what you have written here? – Bruce Nov 08 '22 at 13:34
  • @Bruce I do not agree that it is clear that you know "nothing" about derivatives. The fact that you are asking the question automatically implies that you are learning calculus, and that you have been taught the definition of a derivative in terms of limits. You might also know some "tricks", like the power rule, the chain rule, the product rule, etc. If this is not the case, then you should put more details about your own background into the question. – Xander Henderson Nov 08 '22 at 13:37