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Find the partial derivatives with respect to (a) $x,$ (b) $y$ and (c) $z$.$$f(x, y, z)=4 x^{2}+2 y^{3}+5 z^{5}+3 x-2 y+11 z+12$$

(a) $8 x+3$(b) $6 y^{2}-2$(c) $25 z^{4}+11$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Johns Hopkins University

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Find the partial derivativ…

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Find the first partial der…

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Find the indicated partial…

So if we want to find the partial derivatives with Perspective X, Y and Z, remember, what we're going to do is just assume whichever partial we're taking, all of the other variables are going to be Constance. So if I come over here and first do the partial derivative with respect to X of this, you might see this written as DLF of Dell X or just f sub X. But now what we're going to do is all variables other than axes are going to be assumed to be a constant. So like this. Why term this Z term this Why term this c term, we assume all those air constants. So whenever we take our derivatives, we just treat those like we would a constant in a one variable case. So for X squared, we would use power rules. That gives us eight X. Um so why is a constant Cuba constant so constant? So that whole thing is a constant just plus zero. Same idea here, constant to a power still constant plus zero three x that would just give us three, um, to y will be a constant 11 Z will be a constant and then 12 is a constant. So this was just simplify down 28 x plus three. So then this is going to be our partial with respect to X Now, to get our personal with respect to why so we do del by Dell. Why? And so we end up with Dell F by del y or just Epsom boy. So now we're going to assume the X and disease are all going to be Constance. So when we take those derivatives again, we just treat them like they were constant. So that first term is just gonna be zero plus. So here we use power rule. Eso would be six y squared. This once again will just be a constant constant. I'll give us negative too. And then that's a constant. Then 12 is also a constant. So that would just give us six y squared minus two. So this is our partial with respect to why now to get our partial with respect, to see same thing as we did for the previous two. But now we're going to assume that ex and why will be our Constance So it have DLF by del Z Musical two fz so this first one will just be a constant. So zero next one, also constant zero. Here we use power rule again. So to be 25 times e to the fourth three x and to lie are both going to be constant. 11 z, um is going to just give us 11 and then 12 was a constant. So then that is going to simplify it out 2. 25 z to the fourth US 11. And then that is our partial with respect to see so again, just kind of recap how we would take this derivative. We use all the rules we would have used from just single variable calculus, but we just keep in mind that we assume all other variables will be a constant.

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Find the partial derivatives with respect to (a) $x,$ (b) $y$ and (c) $z$.