Find fx, fy, and fz. f(x, y, z)=y z ln(x y)

Name: Problem 30, Chapter 13: University Calculus: Early Transcendentals
Uploaded: 2020-04-19T15:11:29-08:00
Duration: 5 min 4 s
Channel: Will Erickson
Description: find fx, fy, and fz. f(x, y, z)=y z ln(x y)

Key Concepts

Partial Derivatives

Partial derivatives refer to the differentiation of a multivariable function with respect to one variable while holding the other variables constant. They are fundamental in multivariable calculus for analyzing how a function changes along each coordinate direction independently.

Product Rule

The product rule is a differentiation technique used when a function is expressed as the product of two or more functions. In the context of multivariable functions, when one factor involves the variable of differentiation and another factor is a function of that variable, the product rule helps compute the derivative correctly.

Chain Rule

The chain rule is a method for differentiating composite functions. It is used when a function is applied to another function, allowing one to differentiate the outer function with respect to the inner function, multiplied by the derivative of the inner function with respect to the variable of interest.

Logarithm Differentiation

Logarithm differentiation involves rules for differentiating logarithmic functions, particularly the natural logarithm. This process typically utilizes the property that the derivative of ln(u) with respect to a variable is (u')/u, which is essential when differentiating expressions where the logarithm’s argument is a function of the variable.

Transcript

0:00 Hi there.

00:01 In this problem, we are asked to find the partial derivatives, all three of them, for this function f.

00:08 So let's get started with the partial with respect to x.

00:13 Always remember we're holding y and z constant and treating x as the only variable.

00:19 So we do not actually need a product rule here since y, z are just constants.

00:27 So just as we would normally with one variable, if we're multiplying by a constant, we just keep that constant.

00:32 There.

00:33 So the yz will remain in our answer.

00:35 Now the derivative of log of xy, technically we need a chain rule here since it's log of some inner function of x.

00:43 Now the derivative of log, we know, is one over that inner function.

00:50 And then the chain rule tells us we need to multiply by the derivative of the inner function.

00:55 Now by derivative here, what we mean is partial derivative with respect to x of that inner function.

01:01 So that's the only thing that's different from one variable.

01:04 We just have to make sure we're careful about which variable we're taking the partial derivative with respect to.

01:10 Okay, so we still have the y, z, and we still have the 1 over x, y.

01:16 Now the partial of x, y with respect to x.

01:19 Remember, if we're treating y as a constant, that will just be y.

01:25 And now we can cancel what we can here.

01:27 So we can see the y's will cancel out, and that should be it.

01:31 So we end up with yz over x.

01:38 Okay.

01:42 This next one with respect to y won't be quite as straightforward.

01:45 We'll see y in a second.

01:47 So this time, since we truly do have a function of y, right, it's a function that contains a y, here times a second function that also contains a y.

02:01 We do need the product rule now.

02:04 Neither of these is constant.

02:06 They both have a y in them.

02:07 So let's carefully follow the product rule.

02:10 And just remember, any derivative we take will be a partial with respect to why.

02:14 So remember what the product rule says.

02:16 The derivative is the first function, just as it is.

02:19 Don't change it...