Using Chain Rule to Find Partial Derivatives of a Multivariable Function

[Turbodog66]

Homework Statement

Suppose $$z=x^2 sin(y), x=5t^2-5s^2, y=4st$$
Use the chain rule to find $$\frac{\partial z}{\partial s} \text{ and } \frac{\partial z}{\partial t}$$

Homework Equations

$$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s}$$

The Attempt at a Solution

$$\frac{\partial z}{\partial x} = 2x\sin(y)$$
$$\frac{\partial x}{\partial s} = -10s$$
$$\frac{\partial z}{\partial y} = x^2\cos(y)$$
$$\frac{\partial y}{\partial s} = 4t$$
With those I then substitute the values into the equation, and I came up with $$\frac{\partial z}{\partial s}=2x\sin(y)(-10s)+x^2\cos(y)(4t)=-20(5t^2-5s^2)(s)\sin(4st)+4(5t^2-5s^2)(t)\cos(4st)$$
Where am I going wrong? Also, I am still learning Latex, so I apologize for the crude display.

 

[Ray Vickson]

Homework Statement

Suppose $$z=x^2 sin(y), x=5t^2-5s^2, y=4st$$
Use the chain rule to find $$\frac{\partial z}{\partial s} \text{ and } \frac{\partial z}{\partial t}$$

Homework Equations

$$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s}$$

The Attempt at a Solution

$$\frac{\partial z}{\partial x} = 2x\sin(y)$$
$$\frac{\partial x}{\partial s} = -10s$$
$$\frac{\partial z}{\partial y} = x^2\cos(y)$$
$$\frac{\partial y}{\partial s} = 4t$$
With those I then substitute the values into the equation, and I came up with $$\frac{\partial z}{\partial s}=2x\sin(y)(-10s)+x^2\cos(y)(4t)=-20(5t^2-5s^2)(s)\sin(4st)+4(5t^2-5s^2)(t)\cos(4st)$$
Where am I going wrong? Also, I am still learning Latex, so I apologize for the crude display.

In the last term you should have ##(5t^2-5s^2)^2##, not ##(5t^2-5s^2)##.

 

[Turbodog66]

In the last term you should have ##(5t^2-5s^2)^2##, not ##(5t^2-5s^2)##.

Thanks! Never fails I overlook something simple like that. Other than that, does it appear that I am going about it correctly?