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

$$2 \sqrt{x+y}-\sqrt{x-y}=2 . \text { Find }\left.\frac{d y}{d x}\right|_{(4,0)}$$

$$-1 / 3$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 8

Implicit Differentiation

Derivatives

Missouri State University

Baylor University

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.

01:42

Find $d y$ if $y=\left(2 x…

01:17

$$ \begin{array}{r|c} x &a…

04:28

$$y=x \sqrt{4-x^{2}}, …

01:44

$$y=\sqrt{2 x}, \quad …

for this problem. We have been given a function, and we want to find de y dx at a given point at the 0.0.40 now to find the derivative of this function. This is a great candidate for implicit differentiation because it has excess and wise, all kind of jumbled together. And it's not gonna be very nice or pretty to try to solve this for why. So remember, when we're taking, um, implicit differentiation? It's important to remember that why is a function of X so that if I'm taking, for example, if I had, why squared on? I want to take the derivative of this. Well, I have an exponents. I bring that down so I get to y to the first. But why? It's not just a variable, it's a function. So the chain rule says I then have to take the derivative off that function D y dx. So any time I'm taking the derivative of something with the why, I have to tack that d Y d Exxon because of the change rule, because it's not just a variable, it's standing for a function, even though I haven't written out explicitly what that function is Okay, so let's take a look at our function here. So again, every time I get to the why I'm gonna add on d y d x. One other thing just to kind of review is taking the derivative of a square root. Now, if you can think of a square root as something to the one half power So when I'm taking the derivative here, that one half comes down. So the two goes in the denominator and I end up with an X to the negative one half, which is equivalent of putting that square root in the denominator. So any time we come across a square root, we take the derivative. It puts a two and that square root down in the denominator. Um, just it makes it just easier as we come across those. So I just wanted to review that that step there. Okay, Now let's take our derivative. I have a two taking the derivative of my square root is two times the square root of X plus y. And then in my numerator, I'm gonna multiply by the derivative of what's under that radical derivative of X is one derivative of Why is one D. Y d x? Hey, there's our chain ruling in action. Okay, Next, this could be subtracting and I have another square root again. The two and the radical go in the denominator. The numerator is the derivative of what's under the radical sign. One minus D Y d X, and the derivative of two is just zero. Now we've got two options here. I could go through all of the algebra to solve this for D. Y D X or since that's not really my that's not really my goal here. My goal is to find the value of the derivative at a given point. So I'm going to substitute in foreign zero from my X and y. That's gonna make my algebra a lot simpler. Right? So I have two times one plus D Y d X. That doesn't change. But now here I'll have the square root of four, which is 22 times two is four. Next Fraction one, minus D Y d X denominated. The square root of four is too and again two times two is four equaling zero, and I'm gonna multiply both sides by four. That gets rid of those denominators, so multiply by four multiplied by four. So let's get rid of our parentheses and solve for D Y d X to plus two d Y dx have to distribute that subtraction sign So minus one plus d Y d X equals zero. So that gives me well to minus one is one. I'm gonna move that to the right hand sides. That's negative one. And I have three d Y d x, so de y dx equals negative one third.

View More Answers From This Book

Find Another Textbook

Numerade Educator

01:10

Find two numbers whose sum is 100 such that their product is as large as pos…

03:24

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

01:06

Write the given expression as a sum of logarithms.$$\ln x^{3} y^{4} z^{5…

02:00

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

01:53

Given $f^{\prime}(x)=18 x\left(3 x^{2}+9\right)^{2},$ try to find $f(x)$ by …

01:47

Show that any smooth even function has a relative extremum at its $y$ -inter…

02:02

Sketch the graph of the function defined by the given equation.$$f(x)=5\…

02:23

Sketch the graph of the function defined by the given equation.$$y=f(x)=…

02:36

Find the $x$ -values at which the graph in Exercise 33 crosses its horizonta…

01:45

Write the given expression as a sum of logarithms.$$\log _{4} \frac{x^{1…