The Difference Quotient - Differentiating From First Principles

The gradient of the straight line between two pints is given byWe may generalize this for a curve to find the gradient at any point, but we need to take into account that the curve is actually a curve. In more basic maths we can draw a tangent at a point and find the gradient of the tangent by using the above equation for any two points on the tangent. In practice, when writing down the difference quotient -shown below – we let the distance between the two points tend to zero then the value of the difference quotient at the point is the actual gradient of the tangent at that point and we can writeor

The formula for the difference quotient for a functionat a pointis

Example: Find the difference quotient for the function