How can I use implicit differentiation to find dz/dx and dz/dy in the equation e^(6z) = xyz?

To find the derivatives dz/dx and dz/dy using implicit differentiation from the equation e^(6z) = xyz, we’ll follow these steps:

1. Differentiate both sides: Start with the given equation:

e^(6z) = xyz

2. Apply implicit differentiation to both sides with respect to x and then y separately.
3. Differentiate the left-hand side:

Using the chain rule, differentiate e^(6z):

d/dx(e^(6z)) = e^(6z) * (6 * dz/dx)

4. Differentiate the right-hand side:

Now differentiate xyz:

d/dx(xyz) = yz + x(dy/dx)z + xy(dz/dx)

5. Set the derivatives equal:

So, we have:

e^(6z) * (6 * dz/dx) = yz + x(dy/dx)z + xy(dz/dx)

6. Isolate dz/dx:

Now, isolate dz/dx:

dz/dx(e^(6z) * 6 - xy) = yz + x(dy/dx)z

7. Final formula: Thus:

dz/dx = (yz + x(dy/dx)z)/(e^(6z) * 6 - xy)

This provides you with dz/dx. Next, you can repeat the process for dz/dy:

1. Follow similar steps which include differentiating the original equation implicitly with respect to y.
2. This will yield:

dz/dy = (xz + y(dx/dy)x)/(e^(6z) * 6 - xy)

Hence, dz/dx and dz/dy can be computed using implicit differentiation on the equation e^(6z) = xyz.