Use implicit differentiation to find ∂z/∂x and ∂z/∂y. e^6z = xyz

Solution:

Given e^6z = xyz

To find ∂z/∂x → consider z as a function of x and take y to be a constant

So differentiating with respect to x, we get

e^6z × 6 × ∂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 [6e^6z - xy] = yz

∂z/∂x = yz / [6e^6z - 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^6z × 6 × ∂z/∂y = z × x + xy × 1 × ∂z/∂y

∂z/∂y [6e^6z - xy] = xz

∂z/∂y = xz / [6e^6z - xy]

Therefore, ∂z/∂x = yz / [6e^6z - xy] and ∂z/∂y = xz / [6e^6z - xy]

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Use implicit differentiation to find ∂z/∂x and ∂z/∂y. e^6z = xyz

Summary:

Using implicit differentiation to find ∂z/∂x and ∂z/∂y. e^6z = xyz, we got ∂z/∂x = yz / [6e^6z - xy] and ∂z/∂y = xz / [6e^6z - xy].