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$\begin{array}{l}{\text { Suppose that } f(2)=-3, g(2)=4, f^{\prime}(2)=-2, \text { and }} \\ {g^{\prime}(2)=7 . \text { Find } h^{\prime}(2) .}\end{array}$$\begin{array}{ll}{\text { (a) } h(x)=5 f(x)-4 g(x)} & {\text { (b) } h(x)=f(x) g(x)} \\ {\text { (c) } h(x)=\frac{f(x)}{g(x)}} & {\text { (d) } h(x)=\frac{g(x)}{1+f(x)}}\end{array}$

(a) $h^{\prime}(2)=-38$(b) $h^{\prime}(2)=-29$(c) $h^{\prime}(2)=\frac{13}{16}$(d) $h^{\prime}(2)=-\frac{3}{2}$

Calculus 1 / AB

Chapter 3

Derivatives

Section 4

The Product and Quotient Rules

Missouri State University

Harvey Mudd College

Boston College

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

06:44

Suppose that $$f ( 2 ) = -…

04:07

$\begin{array}{l}{\text { …

03:10

Given $F(2)=1, F^{\prime}(…

01:21

Given $f(x)$ with $f(2)=7$…

08:06

Suppose that $f(4)=2, g(4)…

04:36

Suppose $h=g \circ f .$ Fi…

05:10

Suppose that $ f(4) = 2, g…

00:52

If $g^{\prime}(2)=7$ and $…

01:53

Suppose $F(x)=g(f(x))$ and…

03:42

Find(a) $(f \circ g)(x…

03:06

we are given several pieces of information we have. That f of two is minus three. G of two is for F, Prime of two is minus two and G prime of two is seven. Now we're given several definitions for a function h of X. They're all different functions. We wanted essentially four problems, but in each case we want to find H prime of two. Well, it's not just essentially for different problems. It is for different problems. So the first definition of H of X is five aftereffects minus four g of x. So to find h prime of to the first thing we should do is take the derivative of H, which is five times F prime of X minus four times G prime of X. Because whenever you have a scaler multiplied by a function, when you take the derivative of that, it's what you get is the scaler times the derivative of the function, and you can take derivatives of sums of functions, turn by turn. So that's what we did here and now we just plug into H prime of two. We get five times F Prime of two is minus two minus four times G Prime of two is seven, so you get minus 10 minus 28 equals minus 38. So now that's rewind and use a different function. H this next one is just f of x times gox Now, to take the derivative of halftimes G, we use the product rule which states exactly that the derivative of a product of two functions at times G is of prime of X g of x plus oppa vex G prime of X and now yet again we plug into So f prime of two is minus two g of two is four months, two times four plus off of two is minus three times g Prime of two is seven. So you get my minus 21 equals minus 29 and on to our third function of age Our our third function that we decided we wantto call all of them h in turn. Third function is f of X divided by gox And for this to take the derivative of this, we used the quotient rule which states that would take the derivative of our we take the bottom times the derivative of the top minus the top times the derivative of the bottom all over who's all over the bottom squared. So in this case, when we plug into we get G of two is four times f Prime of two is minus two minus F of two is minus three times G Prime of two is seven all over G of two, which is four squared. So 16 so yet minus eight plus 21 over 16 which is 13 over 16. And now we move on to the last function, which is g of X over one plus FX. So yet again to take the derivative, we use the quotient rule which is gets us the bottom function times the derivative of the top function minus the top function times the derivative of the bottom function. Now the derivative of one class F of X is the derivative of one which is zero plus the derivative of F of axe which is f prime of X, then all over the bottom squared So one plus f of x squared. So now if we plug in to we yet one plus bath of two which is minus three one minus three times g prime of two is seven minus jeep rhyme of Oh, I'm sorry, but the bottom times the drug talk Yes. Oh, there. There is one little mistake here. This should not be a derivative, right? Because it's the bottom that was the derivative of the top, minus the top times the drew to the bottom. Right, So we have one plus f of two is minus three. So one minus three times g prime of two is seven minus G of two is four times f Prime of two is minus two all over one plus f of two. So one minus three. Checking just right is minus two and squared is four. So thing seven minus 14 plus eight over four, which is minus six over four or minus three over two and we're done.

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