# Thread: very simple derivative formula question

1. ## very simple derivative formula question

Theres two derivative formulas being discussed in class, and the one is pretty much completely dominant. That is f(x+h)+f(x)/h

the other is f(b) - f(a)/b-a

my book doesnt really talk much about the second formula, but in its solution manual, it uses it quite a lot without every explaining why thats the preferred choice. While attempting to use the first formula on some problems, I can see why it would be hard, but how does one know before hand which formula to use? Is there a rule of thumb?

thanks

2. ## Re: very simple derivative formula question

Originally Posted by NecroWinter
Theres two derivative formulas being discussed in class, and the one is pretty much completely dominant. That is f(x+h)+f(x)/h

the other is f(b) - f(a)/b-a

my book doesnt really talk much about the second formula, but in its solution manual, it uses it quite a lot without every explaining why thats the preferred choice. While attempting to use the first formula on some problems, I can see why it would be hard, but how does one know before hand which formula to use? Is there a rule of thumb?

thanks
$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

$\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$

the first difference quotient is normally used to find derivatives of functions; the second to find the derivative at a specific value $x = a$ or to determine the differentiability of the function in question at $x = a$. note that both can be used in either role if so desired.

the third difference quotient gives the average rate of change of a function over the interval $[a,b]$

3. ## Re: very simple derivative formula question

Originally Posted by NecroWinter
Theres two derivative formulas being discussed in class, and the one is pretty much completely dominant. That is f(x+h)+f(x)/h

the other is f(b) - f(a)/b-a

my book doesnt really talk much about the second formula, but in its solution manual, it uses it quite a lot without every explaining why thats the preferred choice. While attempting to use the first formula on some problems, I can see why it would be hard, but how does one know before hand which formula to use? Is there a rule of thumb?

thanks
First of all, neither of your derivative formulas are correct.

The first is $\displaystyle f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$, and the second is $\displaystyle f'(c) = \frac{f(b) - f(a)}{b - a}$ for some $\displaystyle c \in [a,b]$.

The second is not actually a formula to evaluate the derivative, it is a theorem that states that a chord between two points on a curve will have the same gradient as the curve itself at some point on the curve in between the chord's two endpoints. This is called the Mean Value Theorem, and it is used extensively in Analysis (in fact, Integral Calculus depends on it...)