z=x^2siny, x=−3s^2+4t^2, y=−4^(st) Find the numerical values of ∂z∂s and ∂z∂t when (s,t)=(3,−4). ∂z∂s(3,−4)= ∂z∂t(3,−4)=

Answer 1

To find the partial derivatives ∂z/∂s and ∂z/∂t at the point (s, t) = (3, -4), you need to use the given expressions for z, x, and y, and apply the chain rule to find these derivatives. Here are the steps:

Given:

z = x^2 * sin(y)

x = -3s^2 + 4t^2

y = -4^(st)

First, calculate ∂z/∂s:

1. ∂z/∂x: Take the derivative of z with respect to x.

∂z/∂x = 2x * sin(y)

2. ∂x/∂s: Take the derivative of x with respect to s.

∂x/∂s = -6s

3. ∂y/∂s: Take the derivative of y with respect to s.

∂y/∂s = -4^(st) * ln(4) * t

Now, apply the chain rule to find ∂z/∂s:

∂z/∂s = (∂z/∂x) * (∂x/∂s) * (∂y/∂s)

∂z/∂s = [2x * sin(y)] * [-6s] * [-4^(st) * ln(4) * t]

Now, plug in the values at (s, t) = (3, -4) into the expression:

∂z/∂s(3, -4) = [2x * sin(y)] * [-6 * 3] * [-4^(3 * -4) * ln(4) * (-4)]

Calculate the values:

∂z/∂s(3, -4) = [2x * sin(y)] * (-18) * (-256 * ln(4) * 4)

Next, calculate ∂z/∂t:

1. ∂z/∂x: We already found this in the first step.

∂z/∂x = 2x * sin(y)

2. ∂x/∂t: Take the derivative of x with respect to t.

∂x/∂t = 8t

3. ∂y/∂t: Take the derivative of y with respect to t.

∂y/∂t = -4^(st) * ln(4) * s

Now, apply the chain rule to find ∂z/∂t:

∂z/∂t = (∂z/∂x) * (∂x/∂t) * (∂y/∂t)

∂z/∂t = [2x * sin(y)] * [8t] * [-4^(st) * ln(4) * s]

Now, plug in the values at (s, t) = (3, -4) into the expression:

∂z/∂t(3, -4) = [2x * sin(y)] * (8 * (-4)) * (-4^(3 * -4) * ln(4) * 3)

Calculate the values:

∂z/∂t(3, -4) = [2x * sin(y)] * (-32) * (-4^(3 * -4) * ln(4) * 3)

Now, calculate both ∂z/∂s and ∂z/∂t using the expressions we derived and the given values of (s, t) = (3, -4).