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The position of a moving object varies with time. The rate at which this position changes at a specific moment is called the instantaneous rate of change.

Mathematically, it's equivalent to the derivative of that function at a point. Derivative is defined as the limit of the difference quotient as the interval approaches zero. It provides the slope of the tangent line to the graph at that point.

For example, consider the function defined by the square root of its input.

To find its derivative, take the square root of the input plus a small increment, subtract the square root of the original input, and divide by the increment.

To simplify further, multiply the numerator and the denominator by the conjugate of the numerator.

This process cancels out the radicals and simplifies into a simpler fractional form, leaving the expression to one divided by twice the square root of the input.

At inputs one and four, the slope is one-half and one-fourth, respectively. But at input zero, the tangent is a vertical line, which means its slope is undefined.