Find fx and ft f(x, t)=((x^2+t^2)/(x^2-t^2))^5

Name: Problem 62, Chapter 6: Calculus and Its Applications
Uploaded: 2021-06-02T15:44:07-08:00
Duration: 12 min 6 s
Channel: Yaw Asomani
Description: Find fx and ft f(x, t)=((x^2+t^2)/(x^2-t^2))^5

Key Concepts

Partial Differentiation

Partial differentiation is the process of finding the derivative of a multivariable function with respect to one variable while keeping the other variables constant. In the context of the given problem, it is used to determine how the function f(x, t) changes with respect to x (f_x) and with respect to t (f_t).

Chain Rule

The chain rule is a fundamental rule in calculus used to compute the derivative of composite functions. It is applicable when a function is composed of another function, as in the given example where f(x, t) is a power of a quotient. The chain rule allows us to differentiate the outer function and then multiply by the derivative of the inner function.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two differentiable functions. In this problem, the expression inside the power is a quotient of two functions of x and t, so the quotient rule is essential for calculating the derivative of that inner function. The rule states that the derivative of a quotient u/v is given by (u'v - uv')/v^2.

Composite Functions

Composite functions are functions that are applied within other functions; meaning one function is nested inside another. In this problem, the function f(x, t) is formed by taking a quotient and then raising it to a power. Understanding composite functions and their differentiation through the chain rule is crucial to solving the problem of finding the partial derivatives.

Transcript

00:01 That's function looks complicated, but if you follow the principles, it's going to be pretty simple, right? i mean, from the outset, it looks sophisticated, but look at a few things, and you can see that it's going to be a little easier to work with.

00:26 Okay, so this is a function.

00:29 You want to find partial derivative with respect to x and partial derivative respect to t.

00:34 Now this is a, first of all, it's a composite function, because it's a function of a function.

00:43 And then the second of all, it is a rational function.

00:47 So it's a combination of the chain rule and the quotient rule, because it's a rational function, something over something, and then it's a composite function as well.

00:56 So we're going to use the chain rule and then the question rule and then we're done.

01:00 So it looks sophisticated, it looks complicated, but if we apply these, core principles is going to be a little easier.

01:10 Now you already know that whenever we're taking a partial derivative with respect to x, we consider anything that has only a t as a constant.

01:19 So in this case, this is a constant and this is a constant, right? so this is going to be a chain rule.

01:28 First of all, we're going to apply the chain rule.

01:30 So this is going to be five.

01:41 Then we take away one from here, which is four.

01:43 Then multiply by the derivative of the inside.

01:46 This is where we're going to apply the quotient rule.

01:50 So the quotient rule is going to have, because we're trying to find the derivative of the inside function.

01:57 So the inside function is a rational function, something over something.

02:02 So the quotient rule is going to apply.

02:04 Okay.

02:05 And the quotient rule has it that we're supposed to make the denominator square, so that is what you have.

02:10 That is a guarantee.

02:11 And the numerator is going to be low to high.

02:14 Going to write the denominator and then take the derivative of the numerator.

02:21 What is the derivative of the numerator? it's just going to be 2x.

02:23 Remember, this t is a constant, so it goes to 0.

02:27 Then plus the numerator is written here.

02:34 Then multiply by the derivative of the denominator.

02:36 What is the derivative of the denominator? it's also going to be 2x.

02:41 I'm a little pressed for space, i'm going to squeeze it in.

02:47 I'm going to do 2x like that, right? so all i got to do is some simplification, which is not so terrible, hopefully.

03:03 So this is going to be five.

03:16 I just rewrote this one.

03:19 Now i want to do something here.

03:22 This is going to be 2x to the third power minus 2xt squared.

03:30 I'm just distributing this 2x to these two terms, right? i'm just foiling, if you will.

03:38 And this is plus 2x to the third power, then plus 2x t squared, right? so that is it for the numerator, and the denominator is just going to be squared.

04:05 So can i do some multiplication? of course i can...

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