## Explicit and Related-Variable Functions ### Worked Example: Composite Trigonometric–Exponential Function Find $$\begin{equation*} \frac{d^2y}{dx^2} \end{equation*}$$ when $$\begin{equation*} y=\tan(e^{3x}). \end{equation*}$$ ### Worked Example: Velocity as a Function of Position If $$\begin{equation*} v=\frac{dx}{dt}, \end{equation*}$$ show that $$\begin{equation*} \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d^2x}{dt^2}. \end{equation*}$$ ## Implicit Functions ### Formula: First Derivative of a General Conic If $$\begin{equation*} ax^2+2hxy+by^2+2gx+2fy+c=0, \end{equation*}$$ then $$\begin{equation*} \frac{dy}{dx} = -\frac{ax+hy+g}{hx+by+f}. \end{equation*}$$ ### Worked Example: Second Derivative of a General Conic If $$\begin{equation*} ax^2+2hxy+by^2+2gx+2fy+c=0, \end{equation*}$$ prove that $$\begin{equation*} \frac{d^2y}{dx^2} = \frac{abc+2fgh-af^2-bg^2-ch^2}{(hx+by+f)^3}. \end{equation*}$$ ## Parametric Functions ### Formula: Second Derivative in Parametric Form Let $$\begin{equation*} x=\phi(t), \qquad y=\psi(t). \end{equation*}$$ If the suffixes denote differentiation with respect to $t$, then $$\begin{equation*} \frac{d^2y}{dx^2} = \frac{x_1y_2-x_2y_1}{x_1^3}. \end{equation*}$$ ### Worked Example: Parametric Second Derivative Let $$\begin{equation*} x=\phi(t), \qquad y=\psi(t). \end{equation*}$$ Show that $$\begin{equation*} \frac{d^2y}{dx^2} = \frac{x_1y_2-x_2y_1}{x_1^3}. \end{equation*}$$ Hence obtain $\dfrac{d^2y}{dx^2}$ when $$\begin{equation*} x=a(\cos\theta+\theta\sin\theta), \qquad y=a(\sin\theta-\theta\cos\theta). \end{equation*}$$ ## Functions of Functions and Inverse Functions ### Formula: Second Derivative of a Composite Function Let $y=f(u)$ and $u=g(x)$. Then $$\begin{equation*} \frac{d^2y}{dx^2} = \frac{d^2u}{dx^2}\frac{dy}{du} + \left(\frac{du}{dx}\right)^2 \frac{d^2y}{du^2}. \end{equation*}$$ ### Worked Example: Composite-Function Second Derivative Let $y=f(u)$ and $u=g(x)$ be differentiable functions. Prove that $$\begin{equation*} \frac{d^2y}{dx^2} = \frac{d^2u}{dx^2}\frac{dy}{du} + \left(\frac{du}{dx}\right)^2 \frac{d^2y}{du^2}. \end{equation*}$$ ### Formula: Second Derivative of an Inverse Function $$\begin{equation*} \frac{d^2x}{dy^2} = -\frac{\dfrac{d^2y}{dx^2}} {\left(\dfrac{dy}{dx}\right)^3}. \end{equation*}$$ ### Formula: Third Derivative of an Inverse Function $$\begin{equation*} \frac{d^3x}{dy^3} = -\frac{ \dfrac{d^3y}{dx^3}\dfrac{dy}{dx} - 3\left(\dfrac{d^2y}{dx^2}\right)^2 }{ \left(\dfrac{dy}{dx}\right)^5 }. \end{equation*}$$ ### Worked Example: Higher Derivatives of an Inverse Function Starting from $$\begin{equation*} \frac{dx}{dy} = \frac{1}{dy/dx}, \end{equation*}$$ obtain formulas for $\dfrac{d^2x}{dy^2}$ and $\dfrac{d^3x}{dy^3}$. ### Worked Example: Second Derivative of a Pedal-Type Function If $$\begin{equation*} p^2=a^2\cos^2\theta+b^2\sin^2\theta, \end{equation*}$$ prove that $$\begin{equation*} p+\frac{d^2p}{d\theta^2} = \frac{a^2b^2}{p^3}. \end{equation*}$$ :::section[Method and Verification] State the domain, select the appropriate derivative formula, carry out the differentiation chronologically, and verify the result by checking a low order or differentiating once more. ::: :::exercise[Practice Questions] 1. Find $d^2y/dx^2$ for $y=\tan(e^{3x})$. 2. If $v=dx/dt$, prove that $d^2x/dt^2=v\,dv/dx$ wherever $v$ is differentiable as a function of $x$. 3. For $x=t^2+1$ and $y=t^3-t$, calculate $d^2y/dx^2$ at $t=1$. ::: :::answer[Answers and Solution Guidance] 1. With $u=e^{3x}$, $y'=3u\sec^2u$ and $y''=9u\sec^2u+18u^2\sec^2u\tan u$. 2. $dv/dt=(dv/dx)(dx/dt)=v\,dv/dx$, and $dv/dt=d^2x/dt^2$. 3. $dy/dx=(3t^2-1)/(2t)$ and $d^2y/dx^2=(3t^2+1)/(4t^3)$, so the value at $t=1$ is $1$. ::: :::faq[Frequently Asked Questions] Q: What is the central method in second-order differentiation? A: The method is to identify the applicable formula or decomposition, state its hypotheses, and then carry out the derivative calculation in the required order. Q: How should a general nth-derivative formula be verified? A: Check one or two initial orders and differentiate the proposed expression once to confirm the formula with $n$ replaced by $n+1$. Q: Which restrictions must be recorded? A: Record every domain condition, excluded denominator value, differentiability requirement, and index restriction used by the formula. ::: :::call-to-action[Continue Learning] subtitle: Complete the practice, verify the main result, and continue to the next repeated-differentiation lesson. button: Next Lesson|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives/partial-fractions-in-repeated-differentiation button-ghost: Review Higher-Order Derivatives|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives :::