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

If $y=x^{3}-\sqrt{2 x},$ find $d x / d t$ when $x=8$ and $d y / d t=64$.

$$256 / 767$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 10

Related Rates

Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

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.

03:11

If $y=\sqrt{3 x^{2}-3},$ f…

01:53

If $L=\sqrt{x^{2}+y^{2}}, …

01:18

Find $d y / d x.$$$y=-…

04:10

Find $d y / d x$ at $x=2$<…

03:27

03:02

If $y^{3}=2 x^{2},$ determ…

02:18

Find $\frac{d x}{d t}$ whe…

02:53

If $x^{2}+y^{2}=25$ and $d…

02:05

for this problem, we've been given an equation Y equals x cubed minus the square root of two X. And our goal is to find DX DT at a certain instance we want when X is eight and d y d t equals 64 now a couple of things to point out before we go to actually solve this problem. First, you can see that both of these derivatives were taken with respect to t. That means when we're finding are derivatives. When I find the derivative of something with an ex, I have to multiply it by d x d t. X In this case is a function of T. Even though I don't know what that function is, it's a function of t. So we need to show that by using the chain rule and multiplying by dx DT the same thing with the why we multiply by d Y d t. When we take the derivative of why Because why is also being treated as a function of t second question that sometimes gets asked, When do we substitute? I know I'm looking at what x equals eight. Can I substitute now before I take the derivative. No, the derivative happens first. So our order is First we find the derivative. Then we can substitute. Okay, Because the derivative is for the general equation. When we substitute, that's a specific instance. So the general derivative first, then we substitute. And the last thing I would point out just because it's showing up in this equation is what happens when we take the derivative of a square root. Well, remember that we can rewrite a square root as, ah, fractional exponents X to the one half. So when I take the derivative of this, okay, that one half comes down. So that puts. There's our one half. We have a two in the denominator. When I subtract one from my exponents, that's extra the negative one half. Well, that puts it down into the denominator and next to the one half or square root of X in the denominator. So just as a reminder of how that looks when we go to take our derivative in a moment, okay, Back to our equation on the left says I just have a y that becomes D y d t. Three x r Sorry x cubed becomes three x squared times, dx, DT There's that chain rule. Now I have a square root So the two and the square root co in the denominator and I multiply that by the derivative of what's under the radical in this case, that's gonna be too again using the chain rule DX DT and I could do a little simplifying two on the top and bottom I can cancel out. Okay, now we're just gonna solve DX. DT is what I want. Everything else. I'm going to get rid off. So d Y d t is 64 x is age, so X squared is going to be 64. There's my DX DT. That's what I'm trying to get, Thio figure out. And here if X is eight, that gives me the square root of 16 in the denominator or it's 4 1/4. Okay, let's simplify this a bit. Well, three times 64 is going to give me, um Well, let's do it this way. I want a common denominator, so I'm actually gonna take this and multiply top and bottom by four. When I do that, I'm going to get on the top 767 over four dx DT when I put these together. So I have 768 minus the one that gives me 767 and I want to get rid of that. So I'm gonna multiply both sides by the reciprocal that gets rid of it on that side. And what I'm left with is DX DT equaling 256 over 767. So that's the derivative of our function. DX DT at are given point.

View More Answers From This Book

Find Another Textbook

Numerade Educator

05:11

Suppose that the oil spill from the damaged hull of a ship forms a circular …

02:15

A cube is measured and each side is found to be 4 inches with a possible err…

03:33

Find the points on the curve whose equation is $x^{2}+y^{2}=16$ nearest and …

03:08

Find those points at which the tangent line to $y=f(x)$ is horizontal for. (…

07:25

A Norman window is one in which the window is constructed by capping a recta…

05:05

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

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

02:14

(a) Find $f^{(3)}(x)$ if $f(x)=1 / x^{2} ;$ (b) Find $s^{\prime \prime}(t)$ …

01:09

For the demand equation $p=(a-b x)^{1 / 2},$ with $a>0$ and $b>0,$ sho…

01:55

An aquarium is to be made in the shape of a rectangular solid with a square …