Derivative power rule conditions

Question

I'm kinda a newbie in calculus, but what are the conditions for the power rule to happen? For example, if we have the number $e$, with the property $\frac{d}{dt}{ {e^t} } = {e^t}$ we can get, using the power rule, that ${t \cdot e^{t-1}} = {e \cdot e^{t-1}}$ and then ${t = e}$. Is there anything wrong? Thanks for your answers!

Answer 1

This is a mistake common to many calculus students, and it is evidence of a lack of fundamentals.

The power rule is used to differentiate powers of functions. These are functions that have some constant in the exponent (e.g. $x^2$, $\sqrt{x-2}$, $\sqrt[7]{3x+1}$, $2x^{0.3}$, etc.).

The power rule cannot be used to differentiate exponential functions. These are functions that have the variable in the exponent (e.g. $2^x$, $\left(\frac12\right)^{x-1}$, $5e^{3x}$, etc.).

As you are just starting to study calculus, whenever you come across a function to differentiate and are not sure whether you can use the power rule, simply ask Is the exponent simply a constant? If the answer is yes, then yes, you can use the power rule. If no, then no.

Answer 2

The power rule only works for functions raised to a power, like x^3, x^4, (x+2)^5, or sqrt(x), etc. The power isn't a variable, it's a constant. When the power is a variable, like e^x, 2^x, we call that an exponential function, and you can't use the power rule to differentiate it. Think about the definition of the derivative, as f(x + h) - f(x) all over h as h goes to zero, and look at what happens for a function like x^2, x^3, x^4 (why does the derivative of x^n become n * x^(n-1)? What happens is interesting to observe), and then do the same for e^x, 2^x, and see how it's fundamentally different. Hope this helps!