Parametric Second Derivatives is a chronological lesson in higher-order differentiation. The source manuscript supplies this topic as part of Methods and Applications of Higher-Order Differentiation, and the presentation below preserves the mathematical intention while making hypotheses explicit. For a parametric curve $$\begin{equation*} x=\phi(t),\qquad y=\psi(t), \end{equation*}$$ the parameter $t$ is not usually the same as $x$. Therefore second derivatives must be computed with care. :::theorem[Second Derivative of a Parametric Curve] Let $x=\phi(t)$ and $y=\psi(t)$ be twice differentiable, and suppose $dx/dt\ne0$. Then $$\begin{equation*} \frac{dy}{dx}=\frac{dy/dt}{dx/dt} \end{equation*}$$ and $$\begin{equation*} \frac{d^2y}{dx^2} = \frac{(dx/dt)(d^2y/dt^2)-(dy/dt)(d^2x/dt^2)}{(dx/dt)^3}. \end{equation*}$$ ::: :::proof Given that $dx/dt\ne0$. By the chain rule, $$\begin{equation*} \frac{dy}{dx}=\frac{dy/dt}{dx/dt}. \end{equation*}$$ Differentiating with respect to $x$ means differentiating with respect to $t$ and then dividing by $dx/dt$. Thus $$\begin{equation*} \frac{d^2y}{dx^2} = \frac{1}{dx/dt}\frac{d}{dt}\left(\frac{dy/dt}{dx/dt}\right). \end{equation*}$$ Using the quotient rule gives the displayed formula. $\square$ ::: :::example[Trigonometric Parametric Curve] Let $x=a\cos t$ and $y=a\sin t$. If $\sin t\ne0$, then $$\begin{equation*} \frac{dy}{dx}=\frac{a\cos t}{-a\sin t}=-\cot t. \end{equation*}$$ Also, $$\begin{equation*} \frac{d^2y}{dx^2}=\frac{-a^2}{(-a\sin t)^3}=\frac{1}{a\sin^3 t}. \end{equation*}$$ ::: --- :::exercise[Practice Questions] 1. State the principal formula or definition used in Parametric Second Derivatives. 2. Apply the main method of this lesson to one simple example of your own choice. 3. Explain one domain, branch, or nonzero-denominator condition that must be checked before using the result. ::: :::answer[Answers and Guidance] 1. Write the formula with all hypotheses before substituting values. 2. Differentiate or simplify step by step, and verify the result by checking a low-order case. 3. The condition must be inherited from the original function or from a division step in the derivation; it must not be added after the computation. ::: :::faq[Frequently Asked Questions] Q: What is the main purpose of this lesson? A: The purpose is to use parametric second derivatives in a controlled way, with all variables, orders, and restrictions stated before calculation. Q: What is the most common error in this topic? A: The most common error is to apply a formula without checking the differentiability, domain, branch, or nonzero-denominator condition required by the formula. Q: How should I verify an answer? A: Differentiate the obtained expression in a low-order case, compare it with the stated formula, and confirm that the excluded values have not been lost. ::: :::call-to-action[Continue Learning] subtitle: Revise the formulae, complete the practice questions, and continue to the next lesson in higher-order differentiation. button: Next Lesson|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives/second-order-composite-functions button-ghost: Review Higher Order Derivatives|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives :::