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If $ g(x) = \sqrt {f(x)}, where Ihe graph of $ f $ is shown, evaluate $ g'(3). $

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$-\frac{\sqrt{2}}{6}$

00:21

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Missouri State University

Harvey Mudd College

Baylor University

University of Nottingham

Lectures

04:40

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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

0:00

If $ g(x) = \sqrt {f(x)}, …

01:40

If $g(x)=\sqrt{f(x)},$ whe…

01:39

Evaluate $f(g(1))$ and $g(…

01:06

00:32

Find the indicated functio…

00:48

00:52

00:22

01:04

00:30

All right. So here we have a graph of F, and you should definitely use the graph shown in your book and not rely on line because mine's just a rough sketch and it might not be quite as accurate. Our goal is to find g prime of three, and we know that G of X is the square root of f of X. I'm going to rewrite G of X as f of X to the 1/2 power, and then we're going to differentiate it using the chain rule. The outer function is the 1/2 power function. So we bring down the 1/2 and we raise the inside to the negative 1/2 power. Then we multiply by the derivative of the inside, and that would be f prime of X. So now we want to find your prime of three. So we substituted three. And for X, it's we've 1/2 times f of three to the negative 1/2 power, tens of crime of three. So we can use the graph to find f of three. So when X equals three, it looks like the graph is at a height of about two. So f of three is too. So now we have 1/2 times to to the negative 1/2 power. That would mean one over square root, too. Now we can also use the graph to find f prime of three. F Prime would be the derivative. So that's the slope of the tangent line and noticed that I have drawn and read the tangent line to the Point X equals three, and it looks like that line has a slope of negative 2/3. You can see the rise of two for the run of three, so we're going to substitute negative 2/3 in there for F prime of three. Now we just need to simplify. And so we have a two on the top of the two on the bottom. We can cancel those, and we have. We're changing our two to the negative 1/2 power and writing it as one over square root, too. So we have one over square root, too. Times negative 1/3. Now, if we don't like to leave a radical in the denominator than we need to take this and multiply it by special form of one square to over square or two, and that's going to give us negative square root, too. Over six

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