3.8E: Exercises
- Page ID
- 116577
This page is a draft and is under active development.
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For exercises 1 - 8, use logarithmic differentiation to find \(\dfrac{dy}{dx}\).
1) \(y=x^{\sqrt{x}}\)
2) \(y=(\sin 2x)^{4x}\)
- Answer
- \(\dfrac{dy}{dx} = (\sin 2x)^{4x}\big[4⋅\ln(\sin 2x)+8x⋅\cot 2x\big]\)
3) \(y=(\ln x)^{\ln x}\)
4) \(y=x^{\log_2x}\)
- Answer
- \(\dfrac{dy}{dx} = x^{\log_2x}⋅\dfrac{2\ln x}{x\ln 2}\)
5) \(y=(x^2−1)^{\ln x}\)
6) \(y=x^{\cot x}\)
- Answer
- \(\dfrac{dy}{dx} = x^{\cot x}⋅\left[−\csc^2x⋅\ln x+\dfrac{\cot x}{x}\right]\)
7) \(y=\dfrac{x+11}{\sqrt[3]{x^2−4}}\)
8) \(y=x^{−1/2}(x^2+3)^{2/3}(3x−4)^4\)
- Answer
- \(\dfrac{dy}{dx} = x^{−1/2}(x^2+3)^{2/3}(3x−4)^4⋅\left[\dfrac{−1}{2x}+\dfrac{4x}{3(x^2+3)}+\dfrac{12}{3x−4}\right]\)
9) Consider the function \(y=x^{1/x}\) for \(x>0.\)
a. Determine the points on the graph where the tangent line is horizontal.
b. Determine the points on the graph where \(y^{\prime}>0\) and those where \(y^{\prime}<0\).
- Answer
- a. \(x=e \approx 2.718\)
b. \(y^{\prime}>0 \text{ for } (0,e)\) and \(y^{\prime}<0 \text{ for } (e,∞).\)