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

Key Concepts

Derivative

The derivative represents the instantaneous rate of change of a function at a specific point. Conceptually, it is the slope of the tangent line to the curve of a function at that point. This idea is central in calculus, and it provides insights into the behavior of functions, such as increasing and decreasing trends.

Difference Quotient

The difference quotient is an expression of the form [f(c+h) - f(c)]/h and is used to compute the derivative by measuring the average rate of change over a small interval. It is the foundational concept that leads to the precise definition of the derivative when the limit as h approaches zero is taken, bridging the gap between discrete changes and continuous change.

Limit

The concept of a limit is critical in calculus as it allows us to understand the behavior of functions as the input values approach a particular point. In the context of derivatives, the limit is used to transition from an average rate of change over an interval to an instantaneous rate of change at a point, thereby defining the derivative formally.

Algebraic Simplification

Algebraic simplification involves manipulating expressions to make them more tractable, especially when computing limits, such as those arising from a difference quotient. Simplification techniques are essential in isolating the variable terms, canceling like terms, and resolving indeterminate forms, ultimately aiding in the evaluation of the limit.

Transcript

00:04 F of x is equal to 8 over x plus 4, we want to find f prime of negative 2.

00:13 Well, we're going to do that by taking the limit as h approach is 0 of this difference quotient.

00:19 F of negative 2 equals the limit as age approaches 0 of f of negative 2 plus h minus f of negative 2 divided by each.

00:33 F of x is 8 over x plus 4.

00:45 F of negative 2 plus h means we're going to substitute negative 2 plus h in for this x.

00:52 So f of negative 2 plus h will be 8 over substituting negative 2 plus h in for x.

01:03 We're going to have negative 2 plus h plus 4.

01:20 So that is f of negative 2 plus h.

01:23 8 over negative 2 plus h plus 4.

01:27 Over negative 2 plus h plus 4.

01:29 Then we need to subtract f of negative 2, meaning we're going to substitute negative 2 now in for x.

01:38 So f of negative 2 will be 8 over negative 2 plus 4.

01:44 So 8 over negative 2 plus 4.

01:52 And then all of that needs to be put over h.

02:01 And we need to take.

02:02 Take the limit of this expression as h approaches 0.

02:08 So first, let's try to simplify things a little bit.

02:14 8 up top here over.

02:18 Now, in the denominator of this fraction, we have negative 2 plus h plus 4.

02:22 So the negative 2 plus 4 is 2.

02:25 So the denominator is really h plus 2.

02:34 And then we're subtracting.

02:36 And then here we have eight up top divided by, well actually let's just make this a lot simpler.

02:51 Here, looking at this fraction, 8 over negative 2 plus 4, negative 2 plus 4 is 2.

02:59 So we really have 8 divided by 2.

03:03 8 divided by 2 is 4.

03:05 So we're really subtracting quantity of 4.

03:11 And then that all has to get put over h, and we need to take the limit of this expression as h approaches zero.

03:36 Now, to help make things easier, let's put this h plus 2 in parentheses.

03:46 Let's put this 4 over 1.

03:48 4 over 1 is still 4.

03:50 Now, what i really want to do is i want to get rid of this h that i'm dividing by.

03:55 So what i'm going to do is i'm going to times this denominator by 1 over h, and i'm going to multiply this numerator by 1 over h...

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