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Find the derivative of the function.$ y = \sqrt \frac {x}{x + 1} $

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00:40

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Missouri State 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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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Find the derivative of the…

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all right. Here we have a composite function and the inside function is a quotient. So we'll be using the chain rule to differentiate. But when we get to the derivative of the inside, we're going to use the quotient rule. And what I would like to do is when I have a square root, I'd like to write it as a 1/2 power. So we have X over X plus one to the 1/2 power. Now let's differentiate. So we're starting with the derivative of the outside. So we're going to bring down the 1/2 and raise a quotient to the negative 1/2 power. Now we're going to find the derivative of the inside. So this is where the quotient rule comes in. So we have the bottom X plus one times the derivative of the top one, minus the top x times the derivative of the bottom one over the bottom squared, X plus one quantity squared. Okay, Now let's see what we can do to simplify this. All right? There are several steps we can take here. First of all, let's work on this numerator. Ignore the times. One that doesn't do anything to us. So what we have is X plus one minus X. So that whole thing is just one. Now, thinking about the negative exponents and negative exponents means a reciprocal. So we can just put that fraction upside down and make it to the positive 1/2 power. So that means we're going to have as our numerator X plus one to the 1/2 power that comes from this part. And then we have a one from this part, and then we have a one from this part. So that's it. And then for the denominator, we have the two that comes from the 1/2. We have X to the 1/2 power that comes from this part. Remember, we took the reciprocal. So now that's on the bottom. And we also have X plus one to the second power, which was already on the bottom. Okay, so the last thing we could do then, is to remember that when we have the same base to a power divided by ah, that same base to a power, we can subtract the powers. Or we could think of it like this. You have an extra three halves of the X plus one quantity on the bottom. So let's cancel the X plus one to the 1/2 on the top of the bottom, leaving us with X plus wanted three halves on the bottom. And then we can also write the X to the 1/2 as a square root. So we have to square root X Times X plus one of the three hats on the bottom, and we just have a one on the top and there's are derivative.

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