Find (d³y)/(dx³) for the implicit function x²+y² = a²

Find $\dfrac{d^{3}y}{dx^{3}}$ for the implicit function $x^{2}+y^{2} = a^{2}$.

Step 1: First derivative

\[ 2x+2yy' = 0 \;\Longrightarrow\; y' = -\frac{x}{y}. \tag{1} \]

Step 2: Second derivative

Quotient rule, then substitute (1): \[y'' = -\frac{y-xy'}{y^{2}} = -\frac{y+\dfrac{x^{2}}{y}}{y^{2}} = -\frac{x^{2}+y^{2}}{y^{3}}.\]

The numerator is the constraint itself, so it is simply $a^{2}$: \[y'' = -\frac{a^{2}}{y^{3}}. \tag{2}\]

Step 3: Third derivative

Since $a$ is constant, (2) needs only the power rule and the chain rule: \[y''' = -a^{2}\cdot\bigl(-3\bigr)y^{-4}y' = \frac{3a^{2}y'}{y^{4}} = \frac{3a^{2}}{y^{4}}\left(-\frac{x}{y}\right).\]

Step 4: Verification against the parametric form

With $x = a\cos t,\ y = a\sin t$, Problem 317 with $b = a$ gives

\[y'''_{xxx} = -\frac{3a\cos t}{a^{3}\sin^{5}t} = -\frac{3\cos t}{a^{2}\sin^{5}t}\]

\[\text{Substituting } \cos t = \frac{x}{a},\ \sin^{5}t = \frac{y^{5}}{a^{5}}:\quad -\frac{3x}{a}\cdot\frac{a^{5}}{a^{2}y^{5}} = -\frac{3a^{2}x}{y^{5}} \;\;\checkmark\]

\[\text{Symbolic check with } a^{2} \to x^{2}+y^{2}:\ \text{residual } 0 \;\;\checkmark\]