Use the Chain Rule to find d z / d t or d w / d t w=x e^y / z, x=t^2, y=1-t, z=1+2 t

Name: Problem 3, Chapter 11: Essential Calculus Early Transcendentals
Uploaded: 2020-05-23T06:57:31-08:00
Duration: 5 min 14 s
Channel: Issa Dababneh
Description: Use the Chain Rule to find d z / d t or d w / d t w=x e^y / z, x=t^2, y=1-t, z=1+2 t

Use the Chain Rule to find d z / d t or d w / d t w=x e^y /...

Key Concepts

Chain Rule in Multivariable Calculus

The chain rule is used to differentiate composite functions, especially when a function depends on several intermediate variables that are themselves functions of another variable. In multivariable calculus, this rule expresses the derivative of a composite function as the sum of the partial derivatives with respect to each variable multiplied by the derivative of that variable with respect to the original independent variable.

Partial Differentiation

Partial differentiation involves taking the derivative of a multivariable function with respect to one variable while keeping the other variables constant. It is essential when applying the chain rule in multivariable settings, as it allows us to compute the sensitivity of the function to changes in each individual variable.

Product Rule

The product rule is a fundamental technique for differentiating functions that are products of two or more functions. When a function is expressed as a product, the derivative is found by differentiating each factor separately and summing the products appropriately, which often occurs in composite functions that involve products of functions of the independent variable.

Differentiation of Exponential Functions

Differentiating exponential functions involves applying the chain rule to handle cases where the exponent itself is a function of the independent variable. This concept is key when dealing with expressions like e^(g(t)), where one must differentiate both the exponential function and the inner function g(t) to obtain the overall derivative.

Transcript

00:01 All right, so in this question, we're asked to determine the derivative of w with respect to t.

00:09 And we're told that x is t squared, y is 1 minus t, and z is 1 plus 2t.

00:16 So x, y, and z are functions of t.

00:19 The first thing we do is we write the chain rule.

00:24 So the derivative of w with respect to t, well, that's just going to be equal to the derivative of w with respect to x.

00:33 Times a derivative of x with respect to t, plus the derivative of w with respect to y by the derivative of y with respect to two, plus the derivative of w respect to z by the derivative of z with respect to t.

00:48 And then now all what we have to do is differentiate and then work out the algebra.

00:55 All right, let's see.

00:55 So let's first determine dw by dx.

01:00 Well, since we're differentiating with respect to x, e to the power of y divided by z can be viewed as a constant.

01:09 So now what's the derivative of x? well, that's just one.

01:12 So for w by dx, we have this term right here, e to the power of y divided by z.

01:20 All right, now what's d x by d t? well, x is just t squared.

01:25 So if we differentiate that with respect to y, that's just two t.

01:29 All right, now what's dw by dy? well, let's think about this.

01:37 So, x can be viewed as a constant.

01:40 Now, what's the derivative of e to the power of y divided by z? well, z can be viewed as a constant, and the derivative of y is just one.

01:54 So then we pull that down, so we get one divided by z, and then we're going to multiply that by e to the power.

02:01 Of y divided by z.

02:03 Again, this is just a chain rule, which we know from before.

02:10 All right, now what's the derivative of y with respect to t? well, y is just 1 minus d...

Please give Ace some feedback