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.$$\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$$

$$\frac{4 x}{9 y}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 8

Implicit Differentiation

Derivatives

Missouri State University

Campbell University

Harvey Mudd College

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.

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.

06:27

Find $d y / d x$ using any…

01:04

Find $d y / d x$. $$y=x^{9…

04:36

05:01

02:28

01:11

Find $d y / d x$.$$

02:05

00:41

Find $d y / d x$ $$y=3 x^{…

01:34

hello

for this problem, we need to find the derivative of X squared over nine minus y squared over four equals one. Now we could do this either explicitly where we solve for y and then take the derivative or implicitly where we leave it, just as it is. And we take the derivative of this function exactly the way we haven't given to us. This is a lesson on implicit differentiation. So I think that's what we're going to do. In this case. The thing to remember with implicit differentiation is that why is a function of X So any time we take the derivative of why we have to use the chain rule, for example, why why squared here? That I have in my function when I take the derivative of this, I have the exponents rule. Exponents comes down. I subtract once it becomes too. Why? But why is a function of X so I have to also multiply it by D Y d X. Even though I don't have explicitly written down what y is in terms of X, we know it is a function of X, so we have to tag that D Y d X on every time we take the derivative of something with a y. Okay, so let's come back to our original equation and see how this plays in when we take the derivative. Okay, first term, this is X. So I don't have to worry about that. I could just straight take the derivative. The two comes down so it becomes two X over nine. Now, here, when I bring down this too is going to be two y over four times. D y d X, right. There's that chain rule and the derivative of one is just zero. Okay, so everything that is not the y dx. I'm gonna move to the other side. So I have negative, and I'm going to simplify this to over four is one half negative y over to D Y d X equals negative two x over nine. Okay, now let's move that negative wire for two over together side. So I have d Y d x. It was negative. Two x over nine times negative to over. Why those negatives cancel on. I just have four x over nine y. So that's the derivative of are given function

View More Answers From This Book

Find Another Textbook

02:15

Determine the extrema on the given interval.$$f(x)=x \sqrt{2-x} \text { …

02:23

Find $d y / d x$ using any method.$$\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$$<…

04:11

The radius of a circle is increasing at the rate of 1 inch per hour. (a) At …

05:05

The area of a square is increasing at the rate of 1 square inch per minute. …

03:07

Let $y=x^{3}-4 x$. (a) Find the equation of the line tangent to this curve a…

03:24

An alternative way to do implicit differentiation is to introduce a mythical…

01:08

A cubic equation of the form $y=a x^{3}+b x^{2}+c x+d$ having three distinct…

01:13

(a) Given $f(x)=x^{2}-4,$ and $g(x)=\left(x^{2}-4\right)^{10} .$ Find the po…

02:51

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

01:36

Determine where the function is concave upward and downward, and list all in…