𝑧
=
𝑒
𝑦
+
2
𝑥
𝑦
+
𝑥
3
−
ln
⁡
𝑥
, find
∂
𝑦
∂
𝑥
when z = c given c is a constant.

Question

𝑧
=
𝑒
𝑦
+
2
𝑥
𝑦
+
𝑥
3
−
ln
⁡
𝑥
 , find 
∂
𝑦
∂
𝑥
 when z = c given c is a constant.

BlackTom AI · Accepted Answer

We begin by analyzing the given relation z = e^y + 2xy + x^3 − ln x with the constraint z = c being constant. This means that as x changes along the level curve z = c, the function z does not change, so dz/dx = 0 along that path.

Next, apply the chain rule for z as a function of x and y, where y is itself a function of x along the level curve. The total derivative with respect to x is:
dz/dx = (∂z/∂x) + (∂z/∂y)·(dy/dx).
Since dz/dx = 0, we have 0 = (∂z/∂x) + (∂z/∂y)·(dy/dx), which can be rearranged to solve for dy/dx:
dy/dx = − (∂z/∂x) / (∂z/∂y).

Compute the partial derivatives from z = e^y + 2xy + x^3 − ln x:
- ∂z/∂x = 2y + 3x^2 − (1/x).
- ∂z/∂y = e^y + 2x.

Substitute these into the formula for dy/dx:
dy/dx = − [2y + 3x^2 − (1/x)] / [e^y + 2x].

A few potential sources of confusion to watch for: (1) ensure the derivative of −ln x is −1/x, not −ln(x) or something similar; (2) remember that e^y differentiates with respect to y, and then you multiply by dy/dx when applying the chain rule; (3) the denominator cannot be zero on the level curve, otherwise dy/dx would be undefined at those points.

Note on the available answer options: in this instance, the provided set of answer choices is empty, so we cannot match a specific option to the derived expression. If a multiple-choice list were given, you would compare each option to the expression above, checking algebraic simplifications and ensuring the sign and terms align with dy/dx = −(2y + 3x^2 − 1/x)/(e^y + 2x).

In summary, the derivative of y with respect to x along the level curve z = c is dy/dx = −(2y + 3x^2 − 1/x) / (e^y + 2x).

𝑧 = 𝑒 𝑦 + 2 𝑥 𝑦 + 𝑥 3 − ln ⁡ 𝑥 , find ∂ 𝑦 ∂ 𝑥 when z = c given c is a constant.Single choice

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