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Find the partial derivatives with respect to (a) $x$ and (b) $y$.$$f(x, y)=3 x^{3}-4 x^{4}+5 x^{3} y^{5}-7 x+9 y^{2}+11$$

$9 x^{2}-16 x^{3}+15 x^{2} y^{5}-7$(b) $25 x^{3} y^{4}+18 y$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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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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Compute the given partial …

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Find all the second partia…

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remember for us to take a partial derivative eso for the first one they want us to do, del by Dell X of this. What we assume is that all the other variables other than the one we're taking the partial of are going to be Constance. So, like this. Why to the fifth this y squared those will just treat like it's any other concept where we're taking this derivative just like we did in single variable calculus. So we would normally write this as f partial X like this. So with the subscript X and now everywhere there's an X. We just take derivative like we wouldn't normally s 03 x cubed. I would give us nine x squared disabuse power rule for X to the fourth negative 16 x cubed power rule again. Now this five executed. Why? To the fifth? So since why is a constant raise it to a power still constant? We can factor that out, and it would be so plus five y to the fifth. And then we take the derivative of just execute. There's going to be three x squared and then negative seven x so they'll just be negative seven Now constant squared times a constant. It's a constant. So if we take the derivative that that should be zero, and then the derivative of 11 is also zero. Now we can go ahead and just clean this up a little bit would be nine x squared, minus 16 X cube. And I'm going to go ahead and multiply the five and three and then move the expert just to the front like that. Um, yes. So this will be our partial here with respect to X. And now, if we were to do the same thing to find our partial but respect to why so we do, Del by Dell. Why on each side and now over here on the left, we have the partial with respect to why which we would write is f sub y. And now all of these X is here. We're going to treat as if those air Constance so a constant cube Time to constant will be constant. So derivative that zero same thing for the next term zero. Now we would have plus five x cubed. And then why? To the Phipps of be five y to the fourth seven x is a constant with respect. Why nine y square Just use power rule. So plus 18 why? And then 11 is also a constant, so I'll just be zero. So you end up with our partial but respect toe. Why is going to be eso? We multiply those together. So 25 x cute. Why to the fourth plus 18? Why And then this is our partial with respect to buy. So again, Just remember when you're taking these partials, whichever partial you're taking all of the other variables or just a constant and you just treat them like you would be taking the derivative in a single variable calculus case.

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