2022-01-19

(y) = In(x^2) at the point where x = e.

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Find the first derivative and evaluate at $x=e$ and $y=2$ to find the slope of the tangent line.Differentiate using the chain rule, which states that $\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]$ is ${f}^{\prime }\left(g\left(x\right)\right){g}^{\prime }\left(x\right)$ where $f\left(z\right)=\mathrm{ln}x$ and $g\left(x\right)={x}^{2}$To apply the Chain Rule, set $u$ as ${x}^{2}$.$\frac{d}{du}\left[\mathrm{ln}\left(u\right)\right]\frac{d}{dx}\left[{x}^{2}\right]$The derivative of $\mathrm{ln}\left(u\right)$ with respect to $u$ is $\frac{1}{u}$.$\frac{1}{u}\frac{d}{dx}\left[{x}^{2}\right]$Replace all occurrences of $u$ with ${x}^{2}$.$\frac{1}{{x}^{2}}\frac{d}{dx}\left[{x}^{2}\right]$Differentiate using the Power Rule.Differentiate using the Power Rule which states that $\frac{d}{dx}\left[{x}^{n}\right]$ is $n{x}^{n-1}$ where $n=2$.$\frac{1}{{x}^{2}}\left(2x\right)$Simplify terms.$\frac{2}{x}$Evaluate the derivative at $x=e$$\frac{2}{e}$