Question

In Exercises \(1-20,\) find \(d y / d x\) by implicit differentiation. $$ x^{2}+y^{2}=36 $$

Short answer

The derivative \(dy/dx\) is \(dy/dx=-x/y\)

Step-by-step solution

Differentiate both sides

Take the derivative of both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating \(y^{2}\), the chain rule is applied, resulting in \(2y*y'\), where \(y'\) stands for \(dy/dx\):\n\(\frac{d}{dx}(x^{2}+y^{2})=\frac{d}{dx}(36)\)

Apply the power rule

Now, apply power differentiation rule which states, the derivative of \(x^{n}\) where n is a constant is \(n*x^{n-1}\), and differentiate \(y^{2}\) using chain rule which gives \(2*y*y'\).\nThis gives: \(2x + 2y*y' = 0\) because the derivative of a constant is 0.

Solve the derivative for \(y'\)

From the equation in Step 2, solve for \(y'\), the derivative of y with respect to x. First, subtract \(2x\) from both sides to isolate the \(y'\) term, then divide by \(2y\) to isolate \(y'\):\n\(y'=-\frac{2x}{2y}=-\frac{x}{y}\).