Question:

Consider the equation of the circle given by the equation: [x^2 + y^2 = r^2] where $r$ is a positive constant.

(a) Find $\frac{{dy}}{{dx}}$ using implicit differentiation.

(b) Find the derivative $\frac{{d^2y}}{{dx^2}}$ using implicit differentiation.

Answer:

(a) To find $\frac{{dy}}{{dx}}$ using implicit differentiation, we will differentiate both sides of the equation with respect to $x$ and apply the chain rule.

Differentiating both sides of the equation $x^2 + y^2 = r^2$ with respect to $x$, we get:

To isolate $\frac{{dy}}{{dx}}$, we rearrange the equation as:

Dividing both sides of the equation by $2y$, we find:

Thus, the derivative of $y$ with respect to $x$ is given by $-\frac{{x}}{{y}}$.

(b) To find the second derivative $\frac{{d^2y}}{{dx^2}}$ using implicit differentiation, we differentiate $\frac{{dy}}{{dx}}$ obtained in part (a) with respect to $x$.

Differentiating $\frac{{dy}}{{dx}}$ = $-\frac{{x}}{{y}}$ with respect to $x$, we get:

Substituting $-\frac{{x}}{{y}}$ for $\frac{{dy}}{{dx}}$ from part (a), we have:

Thus, the second derivative of $y$ with respect to $x$ is given by $\frac{{x}}{{y^2}}$.