2022-01-19

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

xleb123

Skilled2022-01-30Added 94 answers

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

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