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Find $d y / d x$ at the indicated point in two different ways: (a) Solve for $y$ as a function of $x$ and differentiate; (b) Differentiate implicitly.$$x^{2} y+1=2 x; (1,1)$$

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Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 8

Implicit Differentiation

Derivatives

University of Michigan - Ann Arbor

Idaho State University

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.

07:20

Find $d y / d x$ at the in…

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03:16

Carry out the following st…

01:05

Find $d y / d x$ by implic…

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Find $d y / d x$ at the po…

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01:47

Find $d y / d x$ by differ…

01:24

for this problem, we've been given a function X squared. Why? Plus one equals two X. We want to find the derivative of the function and evaluated at the 0.11 Now we're actually going to do this two different ways. Implicit differentiation and explicit differentiation. Explicit means we're going to solve for why and then take the derivative implicit means we're gonna take the derivative even with our X's and y's all jumbled together. So let's start with explicit differentiation because that's what we've been doing up to this point. So we want to solve this for why so non white terms? We're gonna get pushed to the other side and I'm gonna divide by X squared. So there's my why. Now we're going to take the derivative. This is a quotients. We need to use the quotient rule denominator times a derivative of the numerator minus the numerator times the derivative of the denominator all over the denominator squared. And we can pretty this up a little bit. This gives us two X squared minus four X squared plus two X. Just make sure we get all those signs correct, or that's going to be negative. Two X squared plus two x over X to the fourth. Now let's evaluate this at the 0.11 that becomes negative too. Plus 2/1, which has a value of zero. Yeah, okay, let's try implicit differentiation. So I'm going to copy this exact same X squared Y plus one equals two x exact same function. This time, though, we're gonna leave it like this when we have take the derivative of an ex term like two X, we're just going to take the normal derivative. When we take the derivative of something with Why, though, we need to use the chain rule so we would take the derivative of why, like if it's just, let's let's say I was taking the derivative of five y. It's not in this problem, but just as an example, that would be five. That's a derivative of that. But I would have to multiply by D. Y d. X, because why is a function of X even though we don't have it explicitly written down is what that function is. So every time we hit a why, when we're taking that derivative, we're gonna tag on a d y d X Okay, so first X squared, Why, that is a product. So first times, the derivative of the second derivative of why is one de y dx plus the second times the derivative of the first one has a derivative of zero, so that drops off and two X has a derivative of to. Now we're going to solve for D y DX. So I'll move that non de y DX term over to the right hand side. So it's two minus two x y, and I'm going to divide both sides by X squared. So there's my derivative. It has a different form than what we see on on the left hand side. But when we plug everything in, it should give us the same value. So X and Y are both one, so it's gonna be two minus 2/1, and it does both of these, no matter which format we use, both give us a zero

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