Phase-Amplitude Exponential-Trigonometric Products 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. Let $a,b,c\in\mathbb{R}$ and let $$\begin{equation*} r=\sqrt{a^2+b^2}. \end{equation*}$$ If $r>0$, choose the phase $\phi$ by $$\begin{equation*} \cos\phi=\frac{a}{r},\qquad \sin\phi=\frac{b}{r}. \end{equation*}$$ This definition fixes the quadrant of $\phi$. :::theorem[Exponential Cosine Product] Let $y=e^{ax}\cos(bx+c)$ and let $r$ and $\phi$ be defined as above. Then $$\begin{equation*} y^{(n)}=r^n e^{ax}\cos(bx+c+n\phi). \end{equation*}$$ ::: :::proof Given that $y=e^{ax}\cos(bx+c)$. Differentiating once, $$\begin{align*} y' &=ae^{ax}\cos(bx+c)-be^{ax}\sin(bx+c)\\ &=e^{ax}\{a\cos(bx+c)-b\sin(bx+c)\}\\ &=re^{ax}\cos(bx+c+\phi). \end{align*}$$ Repeating the same argument adds $\phi$ to the angle and multiplies the expression by $r$ at each differentiation. Hence, $$\begin{equation*} y^{(n)}=r^n e^{ax}\cos(bx+c+n\phi). \end{equation*}$$ $\square$ ::: :::corollary[Exponential Sine Product] For $y=e^{ax}\sin(bx+c)$, $$\begin{equation*} y^{(n)}=r^n e^{ax}\sin(bx+c+n\phi). \end{equation*}$$ ::: :::example[Negative Exponential Coefficient] Let $y=e^{-x}\cos 2x$. Then $r=\sqrt{5}$ and the phase is determined by $$\begin{equation*} \cos\phi=-\frac{1}{\sqrt5},\qquad \sin\phi=\frac{2}{\sqrt5}. \end{equation*}$$ Therefore $$\begin{equation*} y^{(n)}=5^{n/2}e^{-x}\cos(2x+n\phi). \end{equation*}$$ ::: --- :::exercise[Practice Questions] 1. State the principal formula or definition used in Phase-Amplitude Exponential-Trigonometric Products. 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 phase-amplitude exponential-trigonometric products 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-differentiation button-ghost: Review Higher Order Derivatives|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives :::