Find the first partial derivatives of the function f(r,0) =rln(?+ $') Please ( (a)(b) f(m,y) = J; cos(e) dt...

Find the first partial derivatives of the function. $$h(r, s, t)=e^{r s t}$$

This problem we are asked to find the first partial derivatives of the function W equals E. To the power of the divided by U plus v squared. So we take the partial derivative with respect to the first which is well you don't have to but that's what I'm doing. That will require the application of quotient rule. So we differentiate the numerator and multiply by the denominator unchanged. We have each perevi times U plus v squared. We have minus The derivative of the denominator with respect to be so it's going to be -2 V times the numerator unchanged. So we have minus two ve it's harvey divided by U plus B squared. All squared. Then we want to take the derivative with respect to you which we can do this as its each harvey we're treating that as a constant times the partial derivative with respect to you. D by D U. Of one over U plus V squared. Which makes it a little bit more clear that we are essentially just dealing with a one over type function. And the application of chain rule is very simple here, derivative of you plus B squared with respect to you is just you or just one rather. So we're going to have negative each the power of the divided by U plus V squared squared. For the partial derivative with respect to you

So we have the function w equals e to the t over you plus b squared. And we want to find both partial derivatives we want to find w you on w. V. So let's start off with the partial derivative with respect you. So we have a fraction. So let's use the quote Rule three. So doing that we have the bottom times, the derivative of the top with respect to you. So anytime we're taking derivatives, their partial derivatives with the question roll now. So the bottom times the derivative of the top, um, taking their derivative with respect you eat to the V is just the constant. So the derivative zero minus the top times the derivative of the bottom with respect to you. So I drove over He's worth zero and or a bit of a view is just one. And then we square what's on the bottom So you plus b squared and square that cool thing. So this first term is gone and we end up with negative each of the V over you plus b squared, all squared. And secondly, let's different yet with respect to be so same thing we're gonna do quotient rule. But this time any derivatives are with respect to be and we're treating you as a constant. So we have the bottom times, the derivative of the top with the respective e so narrative of each of the V is each of the B minus the cop times the derivative of the bottom with respect to be this time. So we have to be derivative of you zero. Since it's a constant on we square what's on the bottom? So we have the same denominator because s O. C. Nothing's gonna cancel here. So let's just leave it as you plus B squared you to the V minus to be either the V over you plus B squared quantities word and those are our partial drinking juice.

Actually given a function. And we're asked to find the first partial derivatives of dysfunction function is W equals E T V over U plus v squared. Find the partial derivatives of W. With respect to you. I'll hold the constant and take the derivative with respect to you. Using the question rule, we have the bottom of U plus V squared times the derivative of the top which is a constant. So it's zero minus the top which is E T V has the director of the bottom with respect to you. This is one all over the bottom. New plus V square. I swear this simplifies to negative each the V over useless V square square. He loves for that guy. The guy that gather his life is much better than mine. Find the partial derivatives of W. With respect to V. I'll hold you constant. You take the derivative with respect to be using the question rule. Again, we have the bottom you plus B squared times the dirty at the top, which is again, eats the v minus the top. Each of the times the dirt at the bottom. With respect to be, which is to the all over the bottom. You plus B square square, right? And this simplifies to you plus B squared minus two V. Comes eat the V over you plus b squared squared. That's really good. And you know, I'm rectory, you know, you're really not close.

Were given a function and we're asked to find the first partial derivatives of dysfunction function is F. Of X. Y equals the integral from Y two X of the co sign of me to the T suicide. Well, that was the answer that someone had responded killing every police officer. Oh God, find the partial derivative of F with respect to X. I'll hold Y constant and I'll differentiate with respect to X. Using the funnel meal theorem of calculus. Well, the partial derivative of this integral with respect to X is the co sign of E. To the X. Likewise behind the partial gritty of the death. Respect to why will hold X constant and differentiate with respect to why? Again, using the fundamental theorem. However, this time why is the lower limit for the end of growth? This is the same as then leave partial derivative with respect to why of the opposite of the integral from X to Y. Of the co sign of E. To the T. GT. And therefore are partial derivative is equal to the opposite of the co sign of. Need to the one that really the movie. Yeah.