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).

ECON5005 (ND) Quiz 3 - 2025S1

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

View Explanation

Log in for full answers

Similar Questions

At the given point, find the slope of the curve or the line that is tangent to the curve, as requested. 3x2y - π cos y = 4π, slope at (1, π)

Find dy/dx by implicit differentiation. x1/3 - y1/3 = 1

At the given point, find the line that is normal to the curve at the given point. 6x2y - π cos y = 7π, normal at (1, π)

Use implicit differentiation to find the slope of the tangent line to the curve defined by x y 4 + 6 x y = 5 at the point ( 5 , 1 ) .

Use implicit differentiation to find the slope of the tangent line to the curve defined by x y 2 + 2 x y = 5 at the point ( 3 , 1 ) .

Question at position 1 If 7x2+4y2=17x^2+4y^2=1, then dydx=\frac{dy}{dx}=7x+4y7x+4y4y7x\frac{4y}{7x}14x+8y14x+8y1−14x8y\frac{1-14x}{8y}−7x4y-\frac{7x}{4y}

Question at position 1 If 7x2+4y2=17x^2+4y^2=1, then dydx=\frac{dy}{dx}=7x+4y7x+4y4y7x\frac{4y}{7x}−7x4y-\frac{7x}{4y}1−14x8y\frac{1-14x}{8y}14x+8y14x+8y

If 𝑔 ( 𝑥 ) + 𝑥 cos ⁡ ( 𝑔 ( 𝑥 ) ) = 𝑥 9 , find 𝑔 ′ ( 0 ) .

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler