Math Problem Statement

x * y - cos(x * y) = 1. Use implicit differentiation to find dy/dx.

Solution

To solve this using implicit differentiation, let's proceed step-by-step:

Given: xycos(xy)=1x \cdot y - \cos(x \cdot y) = 1

  1. Differentiate both sides of the equation with respect to xx, applying implicit differentiation to account for yy as a function of xx.

ddx(xy)ddx(cos(xy))=ddx(1)\frac{d}{dx}(x \cdot y) - \frac{d}{dx}\left(\cos(x \cdot y)\right) = \frac{d}{dx}(1)

  1. Applying the product rule to xyx \cdot y:

ddx(xy)=y+xdydx\frac{d}{dx}(x \cdot y) = y + x \cdot \frac{dy}{dx}

  1. Differentiating cos(xy)\cos(x \cdot y) using the chain rule:

ddx(cos(xy))=sin(xy)(y+xdydx)\frac{d}{dx}\left(\cos(x \cdot y)\right) = -\sin(x \cdot y) \cdot \left(y + x \frac{dy}{dx}\right)

  1. Substitute these into the equation:

y+xdydx+sin(xy)(y+xdydx)=0y + x \frac{dy}{dx} + \sin(x \cdot y) \cdot \left(y + x \frac{dy}{dx}\right) = 0

  1. Factor out dydx\frac{dy}{dx} to solve for it:

(x+xsin(xy))dydx=yysin(xy)\left(x + x \sin(x \cdot y)\right) \frac{dy}{dx} = -y - y \sin(x \cdot y)

  1. Finally, solve for dydx\frac{dy}{dx}:

dydx=yysin(xy)x+xsin(xy)\frac{dy}{dx} = \frac{-y - y \sin(x \cdot y)}{x + x \sin(x \cdot y)}

Would you like a deeper explanation or have further questions?

Here are some related questions for further exploration:

  1. How does implicit differentiation differ from explicit differentiation?
  2. Why is the product rule necessary in this problem?
  3. What is the significance of the chain rule in this context?
  4. Can implicit differentiation be used for any type of equation?
  5. How would the solution change if the equation were xy+cos(xy)=1x \cdot y + \cos(x \cdot y) = 1?

Tip: Remember, implicit differentiation is essential when you cannot easily isolate one variable, making it a powerful tool for complex relationships between variables.

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Math Problem Analysis

Mathematical Concepts

Implicit Differentiation
Product Rule
Chain Rule
Trigonometric Differentiation

Formulas

Product rule: d(uv)/dx = u dv/dx + v du/dx
Chain rule for trigonometric functions: d(cos(u))/dx = -sin(u) * du/dx

Theorems

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Suitable Grade Level

Grades 11-12