Finding the Derivative

1. Finding the Derivative The Limit Process

2. 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.

3. 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.

4. A tangent to a graph.

5. 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

6. The slope of the secant line would be (x+h,f(x+h)) (x,f(x)) 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. x+h x

7. 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.

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

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

10. Examples • 1. f(x) = x3 • 2. f(x) = x2 + 2 • Apply the definition of the derivative