Let z x2 sin y where x t2 phi2 and y t2 phi2 evaluate...

Transcript

00:01 Here we are given that z is equal to x squared sine of y and then x is equal to 2s squared plus 2t squared and y is equal to negative 6st.

00:14 We want to find the partial derivative of z with respect to s and the partial derivative of z with respect to t.

00:22 So if we start with s we first are going to take the partial derivative of z with respect to x and multiply it by the partial derivative of x with respect to s.

00:34 Then we are going to add the partial derivative of z with respect to y and multiply that by the partial derivative of y with respect to s.

00:46 So let's go ahead and take the partial derivative of z with respect to x.

00:51 That's just going to be 2x times the sine of y and then we are going to multiply that by the partial derivative of x with respect to s which is going to be 4s.

01:04 Then we are going to add this to the partial derivative of z with respect to y.

01:09 So that is going to be x squared times the cosine of y.

01:13 Then we multiply that by the partial derivative of y with respect to s.

01:18 That is going to be negative 6t.

01:23 So making this a little more simplified we are going to get 8xs times the sine of y plus actually minus 6t x squared times the cosine of y.

01:38 And that is going to be your partial of z with respect to s.

01:41 We are going to repeat the same idea for partial of z with respect to t.

01:47 So we are going to add the partial of z with respect to x times the partial derivative of x with respect to t.

01:53 And then add the partial of z with respect to y times the partial of y with respect to t.

02:01 So we already know what the partial with respect to x is.

02:03 That is 2x times the sine of y and then the partial of x with respect to t is just going to be 4t...

Question

Let $z = x^2 \sin y$ where $x = t^2 - \phi^2$ and $y = t^2 + \phi^2$. Evaluate $\frac{\partial^2 z}{\partial t^2}$ and $\frac{\partial^2 z}{\partial \phi^2}$.

Question

Let $z = x^2 \sin y$ where $x = t^2 - \phi^2$ and $y = t^2 + \phi^2$. Evaluate $\frac{\partial^2 z}{\partial t^2}$ and $\frac{\partial^2 z}{\partial \phi^2}$.

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