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use implicit differentiation to find ∂z/∂x and ∂z/∂y




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e^(3z) = xyz

to find ∂z/∂x → consider z as a function of x and take y to be a constant ... but be careful when you do it b/c it's easy to mess up

so differentiating with respect to x:

e^(3z) * 3 * ∂z/∂x = z * y + xy * 1 * ∂z/∂x ... [using the chain rule on the LHS and the product rule on the RHS]

Factor out the ∂z/∂x:

∂z/∂x [3e^(3z) - xy] = yz

∂z/∂x = yz / [3e^(3z) - xy]

Do the same thing to find ∂z/∂y except consider z to be a function of y and take x to be a constant ...

Differentiating with respect to y:

e^(3z) * 3 * ∂z/∂y = z * x + xy * 1 * ∂z/∂y

∂z/∂y [3e^(3z) - xy] = xz

∂z/∂y = xz / [3e^(3z) - xy]

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