Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Find $d y / d x$ using any method.$$x y=7$$

$-y / x$ or $-7 / x^{2}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 8

Implicit Differentiation

Derivatives

Oregon State University

Baylor University

University of Nottingham

Idaho State University

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.

00:28

Find $d y / d x \text { if…

02:33

Find $d y / d x$ using any…

00:57

Find $\frac{d y}{d x}$.

00:54

01:16

Find $\frac{d y}{d x}$…

00:45

Find $\frac{d y}{d x}$ $y=…

01:06

$$\text {Find } D_{x} y$$ …

00:24

Find $d y / d x$.$$

00:20

Find $y$ given that $x=3$.…

Find $d y$$$y=(7-x)^{8…

for this exercise, we need to take the derivative of X Times y equals seven. Now this is a fairly straightforward, uh, small enough function that it's pretty easy to take the derivative either implicitly or explicitly, And I'm actually going to show both here just because both of them are are pretty straightforward. Let's start with implicit first, because that's what this lessons been about. Remember that with implicit differentiation, whenever we take the derivative of why we're actually needing to invoke the chain rule. Because why is a function of X? So when we take the derivative of something with a why we have to put a d y d Exxon to show that were then taking the derivative with respect to X? Because it's a function of X. So let's see how that plays out in this particular case, X Times Y is a product, so we use the product rule first times the derivative of the second. That's gonna be one times D y DX plus the second time's a derivative of the first, which is just one and the derivative of 70 So let's solve for D Y d X. We'll move the Y over to the right hand side and then divide by X So we get d Y d X equals negative y over X. Now what if we wanted to dio explicit? That means I'm gonna solve for why and then take the derivative while solving this for why gives me seven over x. Another way to think of this is seven times X to the negative one, and that's a little bit easier when it comes to take the derivative. Now let's find D y DX. Bring down that exponents That's negative. Seven. X Subtract one to the negative second. Or I can rewrite this as negative seven over X squared that you can see that thes don't match. They don't look the same however they are, and the way that you get from one to the others will be do a substitution. So let's look at that red one here for a second. It is negative y over X. Well, why is seven over X? Well, we found that right here, So I have negative seven over X for Y over X, and that's negative. Seven over X squared. So they do match. They are the same. It's just a slightly different format because your implicit differentiation often ends up with a Y in there where explicit does not. But even if the forms don't exactly look the same, mathematically, they are still equal.

View More Answers From This Book

Find Another Textbook

05:19

A farmer wishes to build a rectangular pig pen using as little fencing as po…

04:01

The number, $N$ (in million) of VCR's sold in the United States for the…

03:43

Sketch the graph of the function defined in the given exercise. Use all the …

03:23

Use the first derivative to determine where the given function is increasing…

02:51

If the manufacturer of Exercise 4 wishes to guarantee profits of at least $6…

03:55

Find the equation of the tangent line to the curve at the given $x$ -value.<…

02:56

05:05

Classify all critical points.$h(x)=4 x^{3}-13 x^{2}+12 x+9$

01:09

Normally if we say a limit $L$ exists, we mean that $L$ is a finite number I…

02:02

A rectangular page is to contain 96 square inches of print. The top and bott…