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An Introduction to Differentiation and Integration

What is a derivative?

Imagine you have a mathematical function - any function - which I'll call f.
The value of the function depends on some variable, any variable, which I'll call x.
In other words, the value of f (which can also be written as f(x) to reflect its dependence on x) will be different for different values of x.

If f(x) is the function, then df/dx

is the derivative of that function. Basically, dx represents an infinitesimally small (i.e. really small) change in x, and df represents a corresponding change in f(x) .

In other words: If x changes by a certain amount written as dx, then the value of df represents exactly how much f(x) will change as a consequence. So if you change x to x + dx, then f will change to f + df.

Actually, the values of df and dx are too small to write down. But if you divide one by the other, then you get the derivative, which is a ratio, df/dx

, which is equal to the rate of change of f with respect to change in x.

Note that the derivative is not just a simple number - after all, the rate of change of f might not be constant (because only a straight-line graph increases (or decreases) at a constant rate as x increases) - Instead, the derivative is a function, just like f, which depends on x, and gives the instantaneous rate of change of f for any value of x.

x generally represents distance. It doesn't have to be x. It could be a rate of change over time (dt instead of dx). The principle is exactly the same.

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