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

Key Concepts

Derivative

The derivative of a function at a point represents the rate at which the function is changing at that point. It is mathematically defined as the limit of the difference quotient as the increment approaches zero, and it is interpreted as the slope of the tangent line to the function at that point.

Difference Quotient

The difference quotient is an expression that calculates the average rate of change of a function over a small interval. It is given by the formula (f(c+h) - f(c)) / h, and it forms the basis for the definition of the derivative when the limit as h approaches zero is taken.

Limit

The limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a specific value. In the context of finding a derivative, taking the limit of the difference quotient as h approaches zero ensures an exact computation of the instantaneous rate of change of the function.

Transcript

00:04 Okay, f of x is equal to 2x q plus 1 and we want to find f prime of 1 by taking the limited as h approach is 0 of the difference quotient f of 1 plus h minus f of 1 over h.

00:20 So this is going to equal to the limit as h approaches 0 of f of 1 plus h, meaning we're going to substitute 1 plus h into the f of x function everywhere you see x.

00:36 So 2 times 1 plus h cubed and then plus 1.

00:55 So f of x is 2x cubed plus 1, so f of 1 plus h is 2 times 1 plus h cubed plus 1.

01:03 So 2 times 1 plus h cubed plus 1.

01:06 Then we need to subtract f of 1.

01:10 So f evaluated when x is 1, plug 1 in for x, 2 times 1 cubed.

01:22 Plus 1.

01:24 So that is f of 1.

01:30 Then all of that needs to be put over h and we will take the limit of this expression as h approaches 0.

01:38 But first we need to expand this a little bit.

01:41 So 1 plus h cubed expands out to be 1 cubed plus 3 times 1 squared times h.

02:05 So 3 times h.

02:06 So 3 times 3 times 1 squared times h plus 3 times 1 times h squared plus h cute.

02:31 So this portion in the parentheses raised to the third power has the expansion you see up here written in blue.

02:44 And then simplifying what's in this parentheses, 1 cubed is 1, so 2 times 1 is 2 plus 1 is 3.

03:06 So we are taking the limit as h approaches 0.

03:16 Okay, now all of this needs to be times by 2.

03:21 And then we need to add 1, and then we need to subtract this, which is 3.

03:27 So let's put brackets around this 1 plus h cubed equals distance, entire expression written in blue...

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