Differentiate the function by forming the difference...

Key Concepts

Derivative

The derivative of a function represents its instantaneous rate of change at any point. It is obtained by applying the limit process to the difference quotient, providing a precise measurement of how the function behaves locally.

Algebraic Manipulation

Performing algebraic manipulation, such as expanding expressions, combining like terms, and canceling common factors, simplifies the difference quotient. This step is critical to make it feasible to compute the limit as h approaches zero.

Difference Quotient

The difference quotient, expressed as (f(x+h) - f(x))/h, is the fundamental formula used to compute the average rate of change of a function over a small interval. It serves as the foundational concept from which the derivative is defined by measuring how a function changes with an increment h.

Limit Process

The limit process involves taking the value of the difference quotient as the increment h tends to zero, which transitions the average rate of change into the instantaneous rate of change. This limit is crucial for defining the derivative in a rigorous mathematical way.

Transcript

00:01 5x minus x squared and we want to find f prime of x the derivative of f of x by taking the limit of this difference quotient so we need to take the limit as h approach is zero of f of x plus h that means evaluate f at x plus h we're going to plug let me rephrase that.

00:33 We are going to plug x plus h in everywhere you see x.

00:38 So f of x equals 5x minus x squared.

00:42 So f of x plus h is going to equal 5 times x plus h minus x plus h squared.

01:09 So that's f of x plus h, five times x plus h minus x plus h to the second.

01:16 Now we have to subtract f of x.

01:19 Subtracting f of x means subtract 5x minus x squared.

01:34 And all of that has to get put over h.

01:47 Now at this point it looks very complicated, but we just keep doing a little bit of algebra.

01:51 Some things will cancel out, and it'll look a lot better very soon.

01:55 So i'm going to continue this down here.

02:05 It's going to get a little more complicated looking before it gets easier because we actually have to distribute the multiplication by the five.

02:13 We have to take the x plus h and raise it to the second.

02:16 Let's fix that little second power symbol right here.

02:23 Okay.

02:24 So limit as h approach to zero.

02:31 Now, five times x plus h in parentheses is going to be 5x plus 5x plus five times each.

02:43 Now, x plus h squared is going to be x squared plus 2 ,8.

02:53 Hx plus h squared.

03:04 So we need to subtract that entire expression.

03:08 So we're going to subtract each one of these.

03:10 So subtract x squared.

03:33 Okay, so five times x plus h was 5x plus 5h.

03:37 Ah, h plus x plus h to the second comes out to be this little trinomial expression.

03:44 So we need to subtract that entire expression.

03:47 So we're subtracting x squared.

03:49 We just wrote that down.

03:50 Now we need to subtract.

03:52 To 2hx and then we need to subtract the h squared.

04:06 Now we're at the point where we are minusing this expression so minus 5x and then minus a minus x squared is a plus x squared and this all gets put over h.

04:37 This is where things are going to start canceling and start getting a lot better looking.

04:46 So we are taking the limit as h approaches zero.

04:57 Now, 5x minus 5x cancels, minus x squared plus x squared, cancels.

05:06 That leads us with 5h minus 2hx minus h squared over h...

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