Question

Consider the following two equations: Gwz⁴ =sin(y) and yz=−2Gw²

Find ( ∂w/∂z) yusing implicit partial differentiation. Tip: don't do any rearranging before you differentiate, as that can lead to solutions which look very different and confuse STACK. ( ∂w/∂z)y=

Answer 1

To determine ( ∂w/∂z)y using implicit partial differentiation, take the partial derivatives of the given equations with respect to z while treating y as a constant, then solve the resulting system of equations for ( ∂w/∂z)y.

Step-by-step explanation:

To find ( ∂w/∂z)y using implicit partial differentiation for the equations Gwz4 = sin(y) and yz = -2Gw2, we proceed as follows:

First, take the partial derivative of Gwz4 = sin(y) with respect to z, keeping y constant:

4Gwz3(∂w/∂z)y + Gw4 = 0. (Here, sin(y) is treated as a constant because we are differentiating with respect to z and y is held constant.)

Next, take the partial derivative of yz = -2Gw2 with respect to z, again keeping y constant:

y + z(∂y/∂z) = -4Gw(∂w/∂z)y. (Since y is constant with respect to z, ∂y/∂z is 0.)

Thus, y = -4Gw(∂w/∂z)y.

Now we have two equations with two unknowns (∂w/∂z)y appearing in both. We can solve these equations simultaneously to find the desired partial derivative (∂w/∂z)y.