assignment 7, question 4
Implicit Derivative of x² – y² = x + y for Beginners
Mathematical Steps | Plain English Explanation |
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Start with the equation x² – y² = x + y | We have an equation that involves both x and y. We want to find out how y changes as x changes, but y isn’t isolated in the equation. |
Differentiate both sides | To find out how y changes with respect to x, we need to take the derivative of both sides of the equation. This involves a bit of calculus. |
Left Side: 2x – 2y(dy/dx) | The derivative of x² is 2x and the derivative of y² is 2y. But since y is also a function of x, we multiply the derivative of y² by dy/dx. |
Right Side: 1 + dy/dx | The derivative of x is 1 and the derivative of y is dy/dx. We add these together. |
Set them equal: 2x – 2y(dy/dx) = 1 + dy/dx | Now we set the derivatives from both sides equal to each other. This gives us a new equation to solve. |
Isolate dy/dx: dy/dx = (1 – 2x) / (2y – 1) | We rearrange the equation to solve for dy/dx, which tells us how y changes as x changes. |
Final Result
The way y changes with respect to x, also known as dy/dx, for the equation x² – y² = x + y is (1 – 2x) / (2y – 1).