## Problem Statement

Find the derivative.

$y=\frac{e^{\alpha x}(\alpha \cdot \sin \beta x-\beta \cdot \cos \beta x)}{\alpha^{2}+\beta^{2}}$
## Solution

$y^{\prime}=\left(\frac{e^{\alpha x}(\alpha \cdot \sin \beta x-\beta \cdot \cos \beta x)}{\alpha^{2}+\beta^{2}}\right)^{\prime}=\frac{1}{\alpha^{2}+\beta^{2}} \cdot\left(e^{\alpha x}(\alpha \cdot \sin \beta x-\beta \cdot \cos \beta x)\right)^{\prime}=$

$=\frac{1}{\alpha^{2}+\beta^{2}} \cdot\left(\alpha \cdot e^{\alpha x}(\alpha \cdot \sin \beta x-\beta \cdot \cos \beta x)+e^{\alpha x}(\alpha \cdot \cos \beta x \cdot \beta+\beta \cdot \sin \beta x \cdot \beta)\right)=$

$=\frac{e^{\alpha x}}{\alpha^{2}+\beta^{2}} \cdot\left(\alpha^{2} \cdot \sin \beta x-\alpha \cdot \beta \cdot \cos \beta x+\alpha \cdot \beta \cdot \cos \beta x+\beta^{2} \cdot \sin \beta x\right)=$

$=\frac{e^{\alpha x}}{\alpha^{2}+\beta^{2}} \cdot\left(\alpha^{2}+\beta^{2}\right) \cdot \sin \beta x=e^{\alpha x} \cdot \sin \beta x$

## Problem Kuznetsov Differentiation 7-11