Assume that all the given functions have continuous second-order partial derivatives. If z=f(x,

Assume that all the given functions have continuous...

Key Concepts

Chain Rule in Multivariable Calculus

The chain rule in multivariable calculus allows the computation of partial derivatives of a function defined as a composition of other functions. When a function z is expressed in terms of intermediate variables that are themselves functions of other independent variables, the derivative with respect to one of these independent variables is obtained by summing over the partial derivatives with respect to the intermediate variables multiplied by the derivatives of those variables with respect to the independent variable. This rule is essential for transitioning between different coordinate systems or variable sets, keeping track of how changes in one variable affect the composed function.

Mixed Partial Derivatives and Clairaut's Theorem

Mixed partial derivatives involve taking derivatives with respect to two different independent variables in succession. When a function has continuous second-order partial derivatives, Clairaut's theorem states that the mixed partial derivatives are equal regardless of the order of differentiation. This property is crucial in simplifying computations of mixed derivatives in composite functions, as it ensures consistency and allows one to interchange the differentiation order without affecting the result.

Transformation of Variables in Partial Differentiation

Transformation of variables involves rewriting a function in terms of new variables that are functions of the original variables, or vice versa. In the context of partial differentiation, this process requires careful application of the chain rule to relate the derivatives in the new variable space back to those in the original function's variable space. This concept is often used in problems where the function’s natural symmetries or coordinate relationships suggest a change of variables that can simplify the differentiation process.

Transcript

00:01 Alright, so in this video we're asked to determine the derivative with respect to r of dz by d .s.

00:10 So this is basically right here.

00:13 And we're given that z is a function of x and y, and x and y turn our functions of r.

00:20 Well, since that's the case, we have to use the chain rule.

00:24 So first we're going to determine dz by d .s.

00:27 So dz by dz by ds is simply dz by dx times dx by dx.

00:32 Plus d z by d y times d y by d and then the derivative of x with respect to s well x is just r squared plus s so the derivative of this with perspective s is just 2s and the derivative of y with respect to s is just 2 r so we get this right here and now this term since we found d z by d s now we have to find the derivative of this with respect to r so we're going to find the derivative of this with respect to r so we're going to find the derivative of this with respect to r you get this right here.

01:07 And then what do we know from this? well, we have dz by dx times 2s is a product.

01:13 And dz by d .y times 2r is also a product.

01:18 So we're going to have to use the product.

01:20 So first we're going to take the derivative of dz by dx with respect to r and keep 2s constant.

01:26 Then we're going to have to take, then we're going to keep dz by dx constant and take the derivative of 2s with respect to r.

01:35 Plus now we find a derivative of the second term using the product tool we're going to take the derivative of d z by d y with respect to r and keep 2r constant uh yeah and then keep 2r constant and then we're going to keep d z by d y constant and take the derivative with respect to r of 2r and now the only tricky thing the only thing that we need to work on is this term right here so this i'm going to label this one and we're just not all the way and i'm going to label this right here.

02:12 All right.

02:13 These are what we have to try to work on.

02:17 So the derivative with respect to r of dz by dx.

02:20 Again, we're going to use the chain rule.

02:23 So we use the chain rule right here.

02:25 I'm going to use that one more time.

02:27 And then we know that the derivative of x with respect to r is just 2r.

02:31 And the derivative, so the derivative, so this is just 2r...