Power rule vs. Derivative rule I have been learning about derivatives and need some answers. So the power rule is simple you just bring down a power such as $f(x)=x^2$ becomes $f'(x)=2x$. Then with the derivative rule we use the equation:
$$f'(x)=\lim_{h\to 0}\frac{ f(x+h)-f(x)}h$$
I wanted to know if there is a reason to use the derivative rule over the power rule. I understand that is applies mostly to tangent lines but can someone explain it for me
 A: The definition of $f'(x)$ is:
$$f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$
if such a limit exists.
In the particular case when $f(x) = x^n$ where $n$ is an integer, using the definition above, it can be shown that $f'(x) = nx^{n-1}$.
Conceptually, these are two completely different things. One (the "power rule", as you call it) can only be applied to functions of the type $x^n$. The other (which you very strangely name "the derivative rule") can, theoretically, be applied everywhere, but practically, when dealing with functions like polynomials, exponentials and trigonometric functions, the "derivative rule" is never used.
A: The "derivative rule", as you call it, is the definition of the derivative.
The "power rule" is a consequence of applying the above definition to the function $f(x) = x^n$, i.e.,
$$f'(x) = \lim_{h \to 0} \dfrac{(x+h)^n-x^n}h = nx^{n-1}$$
The "power rule", which is a consequence of the definition as mentioned above, is only applicable for $f(x) = x^n$.
There are a lots and lots and lots of functions which are not of the form $x^n$, for instance, $f(x) = \sin(x), \tan^4(x), \log(x), \Gamma(x)$ and so on. For all these functions, we would need to use the definition (or the "derivative rule" in your parlance) to compute the derivative.
A: Historically, mathematicians were trying to find instantaneous rates of change, for example instantaneous speed. What you call the "derivative rule", is the formalization of an incremental method of finding the instantaneous rate of change, ie the derivative.
Typically after a method is developed, short cuts are found from noting common patterns. Your "power rule" is one of the short cuts.
