# Physics with Calculus/Appendix 1/Derivatives

We define a derivative as the limit of a function as the ratio of the changes in the function and the changes in the variable as the change in the variable goes to zero.

i.e.

${\displaystyle {\begin{matrix}{df \over dx}=\lim _{\Delta x\to 0}{{f(x+\Delta x)-f(x)} \over {\Delta x}}\end{matrix}}}$

### An example

Quantity B is found to depend on quantity A such that B is always the square of A.

A ..... B In this case, x=A and f(x) is [the function of x] = B. Look at what happens at the point A=2 (for example).

0 ..... 0 At x=2 f(x)=4. Changing x very slightly, by only 0.01, we have delta x=0.01, so that x+delta x is 2.01.

1 ..... 1 Then the function of (x+delta x) = the function of 2.01 = the square of 2.01 = 4.0401. The difference between f(x+delta x) and f(x)

2 ..... 4 is 4.0401 and 4. It is 0.0401. Dividing it by delta x we have 0.0401/0.01 which is 4.01, meaning that

3 ..... 9 the CHANGE of the function divided by the CHANGE in x is nearly 4 when x=2. If we would have made delta x much smaller

etc. ..... we would have got exactly 4, which happens to be 2 times x at x=2

• In general, if we have a polynomial of x, such as ${\displaystyle x^{3},x^{4},}$ etc then the derivative is as follows:

${\displaystyle f(x)=x^{n};{\frac {df(x)}{dx}}=nx^{n-1}}$

Where ${\displaystyle {\frac {df(x)}{dx}}}$ is the symbol for the derivative of f with respect to x.