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Calculus Level 2

What is the slope of the curve x y x^{y} at x = 1 x=1 ?


The answer is 1.

Prince Loomba by Prince Loomba , 21, India

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1 solution

Richard Costen
Oct 15, 2016

Let y = x x x ∞ = x y y=x^{x^{x^{\infty}}} =x^y .
Take log of both sides, we obtain
ln ⁡ y = ln ⁡ x y ⟹ ln ⁡ y = y ln ⁡ x \ln y=\ln{x^y}\implies \ln y=y\ln x

differentiate both sides implicitly y ′ y = y ′ ln ⁡ x + y x x = 1 ⟹ y = 1 and ln ⁡ x = 0 ∴ y ′ 1 = 0 + 1 1 y ′ = 1 \text{differentiate both sides implicitly}\\ \dfrac{y'}{y}=y'\ln x+\dfrac yx \\ x=1\implies y=1 \text{ and }\ln x=0 \\ \therefore \dfrac{y'}{1}=0+\dfrac11 \\y'=\boxed{1}

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Note: You can use The Latex brackets only around the equations, and not necessarily around text. I've edited the first 2 lines of your solution as an example.

Calvin Lin Staff - 4 years, 7 months ago

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