Find (∂w)/(∂s) if w=4 x+y^2+z^3, x=e^r s^2, y=ln((r+s)/(t)) and z=r s t^2

Name: Problem 246, Chapter 4: Calculus Volume 3
Uploaded: 2019-08-11T12:09:10-08:00
Duration: 6 min 52 s
Channel: Alvar Garcia-Fernandez
Description: Find (∂w)/(∂s) if w=4 x+y^2+z^3, x=e^r s^2, y=ln((r+s)/(t)) and z=r s t^2

step 1 Hello, number eight and welcome back. We're in section 4 .5 of calculus 3. Question 246. Okay, s

Find (∂w)/(∂s) if w=4 x+y^2+z^3, x=e^r s^2, y=ln((r+s)/(t))...

Key Concepts

Partial Derivatives

Partial derivatives measure the rate at which a function changes as only one of its input variables is varied, keeping the remaining variables fixed. This concept is fundamental in multivariable calculus and is essential for understanding how functions behave in higher dimensions.

Chain Rule

The chain rule is a technique for finding the derivative of a composite function, where one function is applied within another. In the context of partial differentiation, it allows the breaking down of complex derivatives into simpler parts by accounting for the dependency of one variable on another.

Function Composition

Function composition refers to the application of one function to the result of another. When dealing with multivariable functions, identifying the nested or composed structure of a function is key to systematically applying differentiation techniques such as the chain rule.

Differentiation of Elementary Functions

Differentiation rules for elementary functions, such as exponential and logarithmic functions, provide the basic building blocks used in calculus. Mastery of these rules is crucial for correctly applying the chain rule and for accurately computing derivatives of more complicated composite functions.

Transcript

00:01 Hello, number eight and welcome back.

00:02 We're in section 4 .5 of calculus 3.

00:06 Question 246.

00:11 Okay, so here we're given the w is the function of x, y, and z.

00:18 It's 4x plus y squared plus z.

00:29 We're also given that x is in turn the function of r and s.

00:36 X is e to the r has squared.

00:45 And y is also a function of rs and t.

00:49 It's the ln of r plus s over t.

01:02 And finally, z is a function of rs and t.

01:07 What is it, rs times t squared? okay, and the question is to find the partial w with respect to s.

01:20 The first partial of w, expect, s.

01:29 Whoops.

01:35 Well, since we have w is a function of x, y, and z, and x, y, and z are in turn functions of rs and t, i think we can do this.

01:44 The first partial w with respect to x times the first partial of x with respect to s right so then it's like the delta x is cancel getting a piece of the change of the first partial w is the first partial s right but then we have to add on the first partial w with respect to y times the first partial of y with respect to s and third let's say on the first partial of w with respect to x i sorry we did x z times the first partial of z with respect to s right so these all simplified the the first partial w to s first partial double respect to s first partial delta x the delta x is cancelled delta x cancelled delta z is cancelled okay so so we get a little bit from this piece, a little bit from this piece, it all adds up to the total differential.

03:08 All right, so here we go.

03:10 What is the first partial of w with respect to x? this is the only x term.

03:15 The derivative 4x is for x is 4.

03:19 All right, what's the first partial of x with respect to s? well, the derivative of e of this stuff is e to this stuff times the derivative of the stuff chain rule, right? elementary calculus chain one.

03:29 And so the derivative of s squared is 2 s...

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