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Find the derivative of y with respect to x for the equation x^2 + y^2 = 25
Published on October 20, 2024
Calculus
The derivative of x² + y² = 25 with respect to x is -x/y.
Question
Find dy/dx if x^2 + y^2 = 25
Answer
Let’s find dy/dx, given that x² + y² = 25.
This is an example of implicit differentiation, where we have ‘y’ tied up with ‘x’ instead of being explicitly defined. Here’s how we tackle it:
1. Differentiate both sides with respect to x:
d(x²)/dx + d(y²)/dx = d(25)/dx
2. Apply the chain rule to the y² term:
2x + 2y(dy/dx) = 0
Remember, when differentiating y² with respect to x, we need to apply the chain rule because y is a function of x.
3. Isolate dy/dx:
2y(dy/dx) = -2x
dy/dx = (-2x) / (2y)
dy/dx = (-2x) / (2y)
4. Simplify:
dy/dx = -x/y
Therefore, the derivative dy/dx of x² + y² = 25 is -x/y.