Power rule vs. Derivative rule

Question

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

Well, the reason to use the definition of derivative is that not every case can be answered with the power rule. — Thomas Andrews, May 02 '15 at 00:04

5xum · Answer 1 · 2015-05-02T00:35:06.593

2

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.

edited May 02 '15 at 00:35

answered May 02 '15 at 00:06

5xum

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Good summary. Maybe note that for -1 that rule doesn't work?(the power rule obviously). Maybe include an example of when using them gives the same answer? – Shinaolord May 02 '15 at 00:23
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@user57404 The power rule does work for $n = -1$. – littleO May 02 '15 at 00:32
@littleO Whoops. I was thinking of integrals, thanks. – Shinaolord May 02 '15 at 00:34

score 1 · Answer 2 · answered May 02 '15 at 00:06

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.

score 0 · Answer 3 · answered May 02 '15 at 00:19

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.

Power rule vs. Derivative rule

3 Answers3