Find f^'(c) by forming the difference quotient...

Key Concepts

Definition of Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the increment approaches zero. It represents the instantaneous rate of change or the slope of the tangent line at that point.

Difference Quotient

The difference quotient is a formula that expresses the average rate of change of a function over an interval. It is calculated by taking the difference between the function’s value at an incremented point and at the original point, divided by the increment, and is central to defining the derivative.

Limit Process

The limit process is used to evaluate the behavior of the function as the increment approaches zero. It resolves the indeterminate form that may appear in the difference quotient and provides the precise value of the derivative, capturing the instantaneous rate of change.

Algebraic Simplification

Algebraic simplification involves manipulating the expression within the difference quotient to cancel common factors and reduce the expression to a form where the limit can be directly evaluated. This step is crucial in effectively computing the derivative from first principles.

Transcript

00:03 Okay, f of x is equal to x squared minus 4x.

00:07 We want to use the limit of the difference quotient to find f prime of 3.

00:14 So f prime of 3 is going to be the limit as h approaches 0 of f evaluated at 3 plus h minus f evaluated at 3, divided by h.

00:40 For this function f of x f prime of 3 will equal the limit of this expression as h approaches 0 so now we just have to go ahead and find out what is f of 3 plus h what is f of 3 substitute those in and then try to do a little bit of algebra and then eventually take the limit f of 3 plus h f of x is x squared minus 4 x so f of 3 plus h is going to be 3 plus h squared, substituting 3 plus h n for x.

01:25 So 3 plus h squared minus 4 times x.

01:32 We're plugging in 3 plus h for x, so minus 4 times 3 plus h.

01:42 And then we need to subtract f of 3.

01:46 So substituting 3 in for x, we're going to subtract, well, f of 3 is going to be 3 squared, minus 4 times straight.

02:03 And that all has to get put over each.

02:13 So f prime of 3 will be the limit of this expression as h approaches 0.

02:29 So expanding this out, 3 plus h squared, is 9 plus 6h plus 8 squared, then minus 4 times 3, minus 12, minus 4x h.

02:41 Minus 3 squared, that subtract 9, and then minus minus 12 is plus 12.

02:47 All over h.

02:49 So our derivative f prime of 3 is going equal to limit of this function as h approaches 0...