Derivative Confusion

Question

I am confused about something.

In derivation we learnt that;

a^x = a^x . lna

Now the question that comes to mind is what is the difference when we have:

a^3 =

Answer 1

If $a$ is a constant, then so is $a^3$, so the derivative of $a^3$ is $0$.

If you are evaluating the derivative of $f(x) = a^x\cdot \ln(x)$ at $x = 3$, then given we know by the rule for such general functions, $f'(x) = a^x\ln(a)$, and to evaluate this at $x = 3$ gives us $$f'(3) = a^3 \ln a$$ which is a constant value, and as soon as we know what $a$ is, we know $f'(3)$.

Answer 2

First, a nitpick: the process of finding a derivative is called differentiation, not derivation.

What you learnt in class was that the derivative of $a^{x}$ with respect to $x$ is $a^{x}\ln(a)$.

So to suggest we apply the result 'when $x=3$' is wrong. We cannot differentiate with respect to $3$ because it is a constant, not a variable. If we differentiate $a^{3}$ with respect to $x$, then we simply get $0$ because $a^{3}$ is constant.

Another note: The usage of $a$ here may be confusing. It has been used as a placeholder for any constant, and should be treated as such. $x$, on the other hand, is understood to take any value for a given $a$ (it is a variable).

Answer 3

$(a^x)'=a^x \ln a$, then it is clear that the differentiation is with respect to $x$. Let's denote this function with $f$, so $f(x)=a^x$. Now, when you take $a^3$, it can be interpreted either as just a number (also considered as a constant function $g(x)=a^3$) or as $f(3)$.