Math Problem Statement

Given that \(2xe^y + 5\cos x = x^3\), find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\).

Solution

To find dydx\frac{dy}{dx} for the given equation 2xey+5cosx=x32xe^y + 5\cos x = x^3, we differentiate both sides of the equation with respect to xx, applying implicit differentiation.

Given:

2xey+5cosx=x32xe^y + 5\cos x = x^3

Steps:

  1. Differentiate each term:

    • For 2xey2xe^y, use the product rule: ddx[2xey]=2ey+2xeydydx\frac{d}{dx}[2xe^y] = 2e^y + 2x e^y \frac{dy}{dx}
    • For 5cosx5\cos x, the derivative is: ddx[5cosx]=5sinx\frac{d}{dx}[5\cos x] = -5\sin x
    • For x3x^3, the derivative is: ddx[x3]=3x2\frac{d}{dx}[x^3] = 3x^2
  2. Combine the results: 2ey+2xeydydx5sinx=3x22e^y + 2x e^y \frac{dy}{dx} - 5\sin x = 3x^2

  3. Isolate dydx\frac{dy}{dx}:

    • Move all terms involving dydx\frac{dy}{dx} to one side: 2xeydydx=3x22ey+5sinx2x e^y \frac{dy}{dx} = 3x^2 - 2e^y + 5\sin x
    • Solve for dydx\frac{dy}{dx}: dydx=3x22ey+5sinx2xey\frac{dy}{dx} = \frac{3x^2 - 2e^y + 5\sin x}{2x e^y}

Final Answer:

dydx=3x22ey+5sinx2xey\frac{dy}{dx} = \frac{3x^2 - 2e^y + 5\sin x}{2x e^y}

Would you like a step-by-step explanation of any part? Let me know!

Here are five related questions to deepen your understanding:

  1. What is implicit differentiation, and why is it necessary here?
  2. How does the product rule apply to 2xey2xe^y?
  3. What is the significance of isolating dydx\frac{dy}{dx} in terms of xx and yy?
  4. How does the chain rule apply when differentiating eye^y?
  5. What happens if x=0x = 0 in the expression for dydx\frac{dy}{dx}?

Tip: Always identify terms that require the chain rule when differentiating implicitly.

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Math Problem Analysis

Mathematical Concepts

Calculus
Implicit Differentiation

Formulas

Product Rule: \(\frac{d}{dx}[uv] = u'v + uv'\)
Chain Rule: \(\frac{d}{dx} e^y = e^y \frac{dy}{dx}\)

Theorems

Differentiation Rules

Suitable Grade Level

Undergraduate Calculus