Finding the Derivative The Limit Process

What is the derivative of something? The derivative of a function f(x) is, mathematically speaking, the slope of the line tangent to f(x) at any point x. It is also called “the instantaneous rate of change” of a function. It can be equated with many real world applications such as; Velocity which is speed such as miles per hour In business, the derivative is called marginal, such as the marginal Cost function, etc.

We call a line which intersects a graph in two points a secant line. It is easy to find the slope of that line. But it takes limits in order to find the slope of a tangent line which touches the graph at only one point. The green line is a secant line because it crosses the blue graph more than once. In particular we focus on the two points illustrated. For animation you need to be connected to the internet.

A tangent to a graph.

To find the slope of the tangent line to a graph f(x) we use the following formula: First let’s see how this formula equates with the slope of a tangent line. Notation: The derivative of f(x) is denoted by the following forms: or

(x,f(x)) (x+h,f(x+h)) x x+h The slope of the secant line would be By decreasing h a little each time we get closer and closer to the slope of the tangent line. By using limits and letting h approach 0 we get the actual slope of the tangent line.

The difference quotient measures the average rate of change of y with respect to x over the interval [x,x+h] In a problem pay attention to that word average. If it is there then you do not use limits.

Example 1: Let Find the derivative f’ of f.

Substitution of numerator. Multiplying out Combining like terms Factor out h Cancel Let h = 0. Done. Find the derivative f’ of f.

Examples 1. f(x) = x 3 2. f(x) = x Apply the definition of the derivative