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\documentclass[12pt]{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{cancel} \usepackage{color} \usepackage{xcolor} \definecolor{my_custom_green}{HTML}{1f8012} \usepackage[utf8]{inputenc} \thispagestyle{empty} \begin{document} \begin{align*} & \text{Let } y = x^{x} \text{, find } \frac{dy}{dx} \\ & {\color{my_custom_green} \text{By using natural logarithm}} \\ & \ln(y) = \ln\left(x^{x}\right) \\ & \ln(y) = x\ln(x) \\ & {\color{blue} \text{By using implicit differentiation}} \\ & \frac{d}{dx} \ln(y) = \frac{d}{dx} x\ln(x) \\ & \frac{1}{y}\frac{dy}{dx} = 1\cdot\ln(x) + \cancel{x}\frac{1}{\cancel{x}} \\ & \frac{dy}{dx} = y(\ln(x) + 1) \\ & {\color{red} \text{Plug back } y \text{ to the equation}} \\ & \boxed{y = x^{x}} \\ & \therefore \frac{dy}{dx} = x^{x}(\ln(x) + 1) \\ \end{align*} \end{document}
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