Differentiation is the process we use to find an expression for the rate of change of a function. This is most easily explained graphically.

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In the graphs above, the green curve shows a function with a point moving along it. The tangent at that (moving) point is shown. The blue curve shows the value of the gradient of that tangent as x changes.

By drawing tangents to a curve we can estimate the gradient at any point but this is time consuming for more than a couple of points, requires a well-drawn graph, and is inaccurate. Differentiation allows us to find a function that describes all of these tangents without having to draw them.

The result of differentiating is the derivative of the function. Because a derivative describes how one value is affected by a change in a variable we say that we find the derivative with respect to that variable. Differentiating the function that gives the green curve above results in the function that gives the blue curve. The blue curve is the gradient function of the green one.

For a graph of y against x the gradient is the derivative of y with respect to x, this is written as \frac{\mathrm{d}y}{\mathrm{d}x}. Using function notation, \frac{\mathrm{d}y}{\mathrm{d}x}=f'(x).

There are numerous applications of derivatives. One of the most easily understood is velocity. Velocity is the rate of change of displacement with respect to time, often written v=\frac{\mathrm{d}x}{\mathrm{d}t}=\dot{x}.