Foundations of Repeated Differentiation is part of the study of higher-order differentiation. The lesson fixes the notation, states the required hypotheses, and then uses the supplied source material in chronological order. :::note[Domain and Notation Convention] For $n\geq 2$, the derivative $f^{(n)}$ is defined recursively from $f^{(n-1)}$. The first derivative $f'$ is the initial case, not another recursive step. ::: :::section[Source Development] The following development follows the uploaded Markdown source for this topic, with editorial correction of notation, domains, and branch conditions where needed. ::: ### Definition: Second-Order Derivative Let $f:I\to\mathbb{R}$ be a differentiable function on an interval $I$. If the derivative $f'$ is differentiable on $I$, then the derivative of $f'$ is called the \textbf{second-order derivative} of $f$. It is denoted by $$\begin{equation*} f''(x),\qquad f^{(2)}(x),\qquad D^2f(x), \qquad \frac{d^2f}{dx^2}. \end{equation*}$$ ### Definition: Higher-Order Derivative Let $n\in\mathbb{N}$ and let $f:I\to\mathbb{R}$ be a function. Suppose that the derivatives $$\begin{equation*} f',f'',f^{(3)},\ldots,f^{(n-1)} \end{equation*}$$ exist on $I$ and that $f^{(n-1)}$ is differentiable on $I$. Then the derivative of $f^{(n-1)}$ is called the \textbf{nth-order derivative} of $f$. It is denoted by $$\begin{equation*} f^{(n)}(x)=D^nf(x)=\frac{d^nf}{dx^n}. \end{equation*}$$ ### Formula: Standard Notations for Successive Derivatives Let $y=f(x)$. Then $$\begin{align*} y'&=f'(x)=Dy=\frac{dy}{dx},\\ y''&=f''(x)=D^2y=\frac{d^2y}{dx^2},\\ y^{(3)}&=f^{(3)}(x)=D^3y=\frac{d^3y}{dx^3},\\ &\ \vdots\\ y^{(n)}&=f^{(n)}(x)=D^ny=\frac{d^ny}{dx^n}. \end{align*}$$ The suffix notation $$\begin{equation*} y_1,\ y_2,\ y_3,\ldots,y_n \end{equation*}$$ may also be used, where $$\begin{equation*} y_r=\frac{d^ry}{dx^r}. \end{equation*}$$ ### Formula: Successive Differentiation For $n\geq 2$, $$\begin{align*} \frac{d^ny}{dx^n} &=\frac{d}{dx}\left(\frac{d^{n-1}y}{dx^{n-1}}\right)\\ &=\frac{d^2}{dx^2}\left(\frac{d^{n-2}y}{dx^{n-2}}\right). \end{align*}$$ ### Lemma: Local Requirement for the Existence of a Higher Derivative Let $n\geq 2$ and let $a$ be an interior point of the domain of a function $f$. If $f^{(n)}(a)$ exists, then $f^{(n-1)}$ must be defined in a neighbourhood of $a$ and differentiable at $a$. Given that $f^{(n)}(a)$ exists. To prove that $f^{(n-1)}$ is defined in a neighbourhood of $a$ and is differentiable at $a$. By the definition of an nth-order derivative, $$\begin{equation*} f^{(n)}(a) = \lim_{h\to 0} \frac{f^{(n-1)}(a+h)-f^{(n-1)}(a)}{h}. \end{equation*}$$ For this limit to be defined, the values $f^{(n-1)}(a+h)$ must exist for all sufficiently small nonzero values of $h$. Therefore, $f^{(n-1)}$ must be defined in a neighbourhood of $a$. The existence of the displayed limit is precisely the differentiability of $f^{(n-1)}$ at $a$. Hence, $f^{(n-1)}$ is defined in a neighbourhood of $a$ and differentiable at $a$. $$\blacksquare$$ ### Remark: Recursive Existence Conditions If $f^{(n)}(a)$ exists, then the corresponding differentiability requirements apply successively to $$\begin{equation*} f^{(n-1)},f^{(n-2)},\ldots,f',f. \end{equation*}$$ Thus, the existence of a higher-order derivative depends on the local existence of all preceding derivatives. --- :::exercise[Practice Questions] 1. For $f(x)=x^5$, compute $f''(x)$ and $f^{(3)}(x)$. 2. State the condition required before $f^{(n)}$ can be defined recursively for $n\geq2$. 3. Explain why $f^{(3)}(x)$ is not the cube of $f(x)$. ::: :::answer[Answers and Guidance] 1. $f''(x)=20x^3$ and $f^{(3)}(x)=60x^2$. 2. The derivative $f^{(n-1)}$ must exist on the interval and be differentiable at the point or on the interval under discussion. 3. $f^{(3)}$ denotes the third derivative, whereas $(f(x))^3$ denotes a power of the function value. ::: :::faq[Frequently Asked Questions] Q: What is the main purpose of this lesson? A: The purpose is to use foundations of repeated differentiation 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, 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. ::: :::review-history[Academic Review Notes] - Source mathematics reviewed against the uploaded higher-order differentiation manuscript. - Domains, branch conventions, exercises, answers, FAQ formatting, and final call-to-action placement checked. ::: :::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/standard-nth-derivatives button-ghost: Review Higher Order Derivatives | /real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives :::