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Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ f(x) = x^{3/2} $

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$\frac{3}{2} x^{1 / 2}$

05:42

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Oregon State University

Harvey Mudd College

Idaho State University

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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So in this problem were given this function F of X is X to the To the 3/2. And were asked to do several things. Were asked to find the domain of this, the derivative of using the function and the domain of the derivative. So, I want to start with the domain here, first of all. So the domain is 02 infinity. As I cannot do a negative number To the 1/2 power. Okay. So there's my domain for my function. Now, definition of derivative definition derivative says F prime of X is equal to the limit as H goes to zero of F of X plus H minus F of X over H. All right. So for us then that means derogative fx is the limit as H goes to zero of X plus H two, the three house minus X to the three house all over H. And we can see real quick is that if we multiply this by X plus H two the three halves plus X to the three house over X plus H three halves plus X to the three halves. Right? Because what are we doing? We're doing a let's see, we're doing a minus B times A plus B equals a squared minus b squared. All right. So that means this is the limit. His page goes to zero of x plus age to the three halves squared. Well, the one half power squared cancels out. And so I'm left with X plus H cubed minus X cubed, aren't I? Because extra three halves squared would be Next to the 3rd. Okay. Age times X plus H. 23 halves plus X. To the three halves. All right now multiply. This out Limit is H goes to zero. Uh X cubed plus three X square at H plus three X. H squared plus H cubed minus X cubed over. H times X plus H. Two. The three halves Plus X. 2 3/2. All right. And what do we notice? First of all we noticed that X cubed minus X cubed. So that's gone next. We noticed that I got an H. Here in the denominator and H. And that term and H there and I can take one of those ages cancel the H. Is out. Okay And so I'm left with what I'm left with the limit as H goes to zero of three x squared Plus three XH plus H squared over X plus H. Two the three halves plus X to the three halves. Okay now perform the limit Well when h goes to zero that term goes to zero that term goes to zero and that H goes to zero doesn't it? Some left with three X squared over execute was X cubed. Extra three halves plus extra three halves. That's two X. to the three. Well I didn't want to write very well let's try that again. X. to the three house. So that's three over to well X squared over extra three halves. That's X. To the two minus three halves in the export up there. So that leaves me X to the one half, doesn't it? Okay. And the domain then is still zero to infinity. Just like we said above. Right, We can't do a negative number to the one half, unless we can't do the square root of a negative number. So here is my derivative and its domain.

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