Derivative of a variable power f(x)^g(x) by taking logarithms

1. A.
   $(\sin x)^{\tan x}[1+\sec^2 x\cdot\log(\sin x)]$
2. B.
   $\tan x\cdot(\sin x)^{\tan x-1}\cdot\cos x$
3. C.
   $(\sin x)^{\tan x}\cdot\sec^2 x\cdot\log\sin x$
4. D.
   $\tan x\cdot(\sin x)^{\tan x-1}$

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1. A.
   $\ x^{x}(1 + \log x) + x^{\frac{1}{x}}\frac{1}{x^{2}}(1 - \log x)$
2. B.
   $\left( x^{x} + x^{\frac{1}{x}} \right)\left\lbrack 1 + \log x + \frac{1}{x^{2}}(1 - \log x) \right\rbrack$
3. C.
   $\left( x^{x} + x^{\frac{1}{x}} \right)\left\lbrack (1 + \log x) - \frac{1}{x^{2}}(1 - \log x) \right\rbrack$
4. D.
   $x^{x}(1 + \log x) - x^{\frac{1}{x}}\frac{1}{x^{2}}(1 - \log x)$

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1. A.
   $\ - 2(1 + \log2)$
2. B.
   $2(1 + \log2)$
3. C.
   $2 + \log2$
4. D.
   $1 + \log2$

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1. A.
   $\ x^{\left( x^{x} \right)}\left( x^{x} + 1 + \log x \right)$
2. B.
   $\ x^{\left( x^{x} \right)}\left( x^{x} + \log x \right)$
3. C.
   $\ x^{\left( x^{x} \right)}\left( x^{x} + x^{x - 1}\log x(1 + \log x) \right)$
4. D.
   $\ x^{\left( x^{x} \right)}\left( x^{x - 1} + x^{x}\log x(1 + \log x) \right)$

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1. A.
   increasing
2. B.
   strictly increasing.
3. C.
   decreasing
4. D.
   strictly decreasing

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1. A.
   $2^x\log 2 + (\sin x)^x[x\cot x + \log(\sin x)]$
2. B.
   $2^x\log 2 + (\sin x)^x[x\cot x - \log(\sin x)]$
3. C.
   $2^x\log 2 + (\sin x)^x[-x\cot x - \log(\sin x)]$
4. D.
   $2^x\log 2 + (\sin x)^x[-x\cot x + \log(\sin x)]$

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1. A.
   $-\dfrac{(y + x\log y)}{(x + y\log x)} \cdot \dfrac{y}{x}$
2. B.
   $\dfrac{(y + x\log y)}{(x + y\log x)} \cdot \dfrac{y}{x}$
3. C.
   $-\dfrac{(x + y\log y)}{(y + x\log x)} \cdot \dfrac{y}{x}$
4. D.
   $\dfrac{y}{x}$

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1. A.
   $\dfrac{y}{2\sqrt{x}}(2 + \log x)$
2. B.
   $\dfrac{y}{\sqrt{x}}(2 + \log x)$
3. C.
   $\dfrac{y}{2\sqrt{x}}(2 - \log x)$
4. D.
   $\dfrac{y}{\sqrt{x}}(2 - \log x)$

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1. A.
   $\dfrac{y^2(1 + \log y)}{x^2(1 - \log x)}$
2. B.
   $\dfrac{y^2(1 + \log x)}{x^2(1 - \log y)}$
3. C.
   $\dfrac{-y^2(1 - \log x)}{x^2(1 - \log y)}$
4. D.
   $\dfrac{-x^2(1 - \log x)}{y^2(1 - \log y)}$

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1. A.
   $(\log x)^x$
2. B.
   $\log(\log x)$
3. C.
   $x^{\log x}$
4. D.
   $\log(\log x^e)$

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