Find the gradient ∇f. f(x, y)=sin^3(x^2 y)

Name: Problem 6, Chapter 12: Calculus
Uploaded: 2021-06-29T21:37:33-08:00
Duration: 4 min 56 s
Channel: Alex Roush
Description: Find the gradient ∇f. f(x, y)=sin^3(x^2 y)

Key Concepts

Gradient

The gradient of a function is a vector composed of its first-order partial derivatives with respect to each variable. It represents both the direction and the rate of the fastest increase of the function, and is fundamental in optimization and analysis of multivariable functions.

Partial Derivatives

Partial derivatives involve differentiating a multivariable function with respect to one variable while treating the other variables as constants. They provide the components necessary for constructing the gradient vector, and are key to understanding changes in functions with several variables.

Chain Rule for Multivariable Functions

The chain rule in the context of multivariable functions is used to differentiate composite functions. It allows one to differentiate a function whose argument itself is a function of other variables by multiplying the derivative of the outer function with respect to the inner function by the derivative of the inner function with respect to the variable of interest.

Composite Functions

A composite function is a function formed when one function is applied to the result of another function. In differentiation, recognizing composite functions is essential since it requires the application of the chain rule to correctly compute derivatives, especially when dealing with nested functional forms.

Transcript

00:01 Okay, so we were to find the gradient of this function, and recall that the gradient is just the, as a vector with the partial derivatives as components.

00:09 So we're going to take partial f to x, partial f with respect to y.

00:13 First thing i'm going to do, since we're taking derivatives, i'm actually going to write this slightly differently because this three doesn't mean what a lot of people think it thinks it means.

00:21 So this is sign of x squared y, and this quantity is being cubed or multiplied by itself three times.

00:30 So that's just something to keep in mind as we're doing this.

00:33 Okay, so the partial with respect to x of this function, the way that you do it is you essentially treat y as a constant.

00:48 And if you do that, the first thing you're going to look at is the overarching function, which in this case is something being raised to the third power.

00:57 So the power rule says that you would bring that three down and you would multiply at times this function being raised to the second power times the derivative of the inside function.

01:14 Okay? so this is the chain rule.

01:17 So the derivative of the inside function, well, again, the overarching function is sine of something, and the derivative of sine is just cosine.

01:29 But then we have to take the derivative of, now we're here.

01:33 This is the inside function now.

01:34 So the derivative of x squared y, well, y is a constant, so that can just be moved kind of out in front, and the derivative of x squared with respect to x is now, now 2x.

01:47 So that's what we get.

01:49 So if we wanted to clean it up just a little bit, we could write this as, well, we have a 3 right here, so this would be 3 times 2 is 6.

01:58 So i'll write 6, x, x, y, sine squared of x squared y, cosine of x squared y.

02:14 So that's just one component...

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