## Powers, Reciprocals, and Logarithms ### Formula: Nth Derivative of a Real Power Let $k\in\mathbb{R}$ and let the required derivatives exist. Then $$\begin{equation*} \frac{d^n}{dx^n}(x^k) = k(k-1)(k-2)\cdots(k-n+1)x^{k-n}. \end{equation*}$$ ### Formula: Nth Derivative of a Positive Integral Power For $k\in\mathbb{N}$, $$\begin{equation*} \frac{d^k}{dx^k}(x^k)=k!, \qquad \frac{d^n}{dx^n}(x^k)=0 \quad\text{for }n>k. \end{equation*}$$ ### Formula: Nth Derivative of a Negative Power For $k>0$, $$\begin{equation*} \frac{d^n}{dx^n}(x^{-k}) = (-1)^n \frac{k(k+1)\cdots(k+n-1)}{x^{n+k}}. \end{equation*}$$ For $k\in\mathbb{N}$, $$\begin{equation*} \frac{d^n}{dx^n}(x^{-k}) = (-1)^n \frac{(k+n-1)!}{(k-1)!\,x^{n+k}}. \end{equation*}$$ ### Formula: Nth Derivative of a Reciprocal $$\begin{equation*} \frac{d^n}{dx^n}\left(\frac{1}{x}\right) = \frac{(-1)^n n!}{x^{n+1}}. \end{equation*}$$ ### Formula: Nth Derivative of a Logarithm For $x>0$, $$\begin{equation*} \frac{d^n}{dx^n}(\log x) = \frac{(-1)^{n-1}(n-1)!}{x^n}. \end{equation*}$$ ### Formula: Nth Derivative of a Shifted Reciprocal $$\begin{equation*} \frac{d^n}{dx^n}\left(\frac{1}{x-a}\right) = \frac{(-1)^n n!}{(x-a)^{n+1}}. \end{equation*}$$ ### Formula: Nth Derivative of a Linear Reciprocal $$\begin{equation*} \frac{d^n}{dx^n}\left(\frac{1}{ax+b}\right) = \frac{(-1)^n n!a^n}{(ax+b)^{n+1}}. \end{equation*}$$ ## Quadratic Denominators ### Worked Example: Difference of Two Squares If $$\begin{equation*} y=\frac{1}{x^2-a^2}, \end{equation*}$$ find $y_n$. ### Formula: Nth Derivative of a Difference-of-Squares Reciprocal $$\begin{equation*} \frac{d^n}{dx^n}\left(\frac{1}{x^2-a^2}\right) = \frac{(-1)^n n!}{2a} \left[ \frac{1}{(x-a)^{n+1}} - \frac{1}{(x+a)^{n+1}} \right]. \end{equation*}$$ ### Worked Example: Sum of Two Squares If $$\begin{equation*} y=\frac{1}{x^2+a^2}, \end{equation*}$$ find $y_n$. ### Formula: Nth Derivative of a Sum-of-Squares Reciprocal Let $x=r\cos\theta$, $a=r\sin\theta$, and $r=\sqrt{x^2+a^2}$. Then $$\begin{equation*} \frac{d^n}{dx^n}\left(\frac{1}{x^2+a^2}\right) = \frac{(-1)^n n!}{a\,r^{n+1}}\sin((n+1)\theta). \end{equation*}$$ Equivalently, $$\begin{equation*} \frac{d^n}{dx^n}\left(\frac{1}{x^2+a^2}\right) = \frac{(-1)^n n!}{a^{n+2}} \sin^{n+1}\theta\sin((n+1)\theta). \end{equation*}$$ ### Corollary: Nth Derivative of the Inverse Tangent If $$\begin{equation*} y=\tan^{-1}\left(\frac{x}{a}\right), \end{equation*}$$ then $$\begin{equation*} y_n = \frac{(-1)^{n-1}(n-1)!}{a^n} \sin^n\theta\sin(n\theta), \qquad \theta=\tan^{-1}\left(\frac{a}{x}\right). \end{equation*}$$ ### Worked Example: Rational Function with a Positive Quadratic Let $ac>b^2$. Prove that $$\begin{equation*} D^n\left(\frac{b+cx}{a+2bx+cx^2}\right) = (-1)^n n! \left(\frac{c}{a+2bx+cx^2}\right)^{(n+1)/2} \cos\left[ (n+1)\tan^{-1} \left( \frac{\sqrt{ac-b^2}}{b+cx} \right) \right]. \end{equation*}$$ ## Exponential and Trigonometric Functions ### Formula: Nth Derivative of an Exponential Function $$\begin{equation*} D^n(e^{ax})=a^ne^{ax}. \end{equation*}$$ ### Formula: Nth Derivative of a Sine Function $$\begin{equation*} D^n(\sin ax) = a^n\sin\left(ax+\frac{n\pi}{2}\right). \end{equation*}$$ ### Formula: Even-Order Derivative of a Sine Function $$\begin{equation*} D^{2n}(\sin ax)=(-a^2)^n\sin ax. \end{equation*}$$ ### Formula: Nth Derivative of a Cosine Function $$\begin{equation*} D^n(\cos ax) = a^n\cos\left(ax+\frac{n\pi}{2}\right). \end{equation*}$$ ### Formula: Even-Order Derivative of a Cosine Function $$\begin{equation*} D^{2n}(\cos ax)=(-a^2)^n\cos ax. \end{equation*}$$ ### Formula: Exponential–Cosine Product Let $a>0$, $r=\sqrt{a^2+b^2}$, $a=r\cos\phi$, and $b=r\sin\phi$. Then $$\begin{equation*} D^n\left(e^{ax}\cos(bx+c)\right) = r^ne^{ax}\cos(bx+c+n\phi). \end{equation*}$$ ### Formula: Exponential–Sine Product Let $a>0$, $r=\sqrt{a^2+b^2}$, $a=r\cos\phi$, and $b=r\sin\phi$. Then $$\begin{equation*} D^n\left(e^{ax}\sin(bx+c)\right) = r^ne^{ax}\sin(bx+c+n\phi). \end{equation*}$$ ### Formula: Negative Exponential–Cosine Product If $-a=r\cos\phi$ and $b=r\sin\phi$, then $$\begin{equation*} D^n\left(e^{-ax}\cos(bx+c)\right) = r^ne^{-ax}\cos(bx+c+n\phi). \end{equation*}$$ ### Worked Example: Sum of Sine and Cosine If $$\begin{equation*} y=\sin kx+\cos kx, \end{equation*}$$ prove that $$\begin{equation*} y_n = k^n\left\{1+(-1)^n\sin 2kx\right\}^{1/2}. \end{equation*}$$ ### Worked Example: Exponential Function Multiplied by a Squared Cosine If $a,b>0$ and $$\begin{equation*} y=e^{ax}\cos^2 bx, \end{equation*}$$ find $y_n$. :::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 $D^n(x^{-3})$ for $x\ne0$. 2. Find $D^n(\log x)$ for $x>0$. 3. Let $r=\sqrt{a^2+b^2}$ and choose $\phi$ such that $a=r\cos\phi$ and $b=r\sin\phi$. Find $D^n(e^{ax}\cos bx)$. ::: :::answer[Answers and Solution Guidance] 1. $D^n(x^{-3})=(-1)^n\dfrac{(n+2)!}{2x^{n+3}}$. 2. $D^n(\log x)=(-1)^{n-1}\dfrac{(n-1)!}{x^n}$. 3. $D^n(e^{ax}\cos bx)=r^n e^{ax}\cos(bx+n\phi)$. ::: :::faq[Frequently Asked Questions] Q: What is the central method in standard nth derivatives? 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/second-order-differentiation button-ghost: Review Higher-Order Derivatives|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives :::