For , let . Which of the following is an expression for…

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

For x>0x>0, let p(x)=e2x ln⁡(5x)log⁡3xp(x)=\dfrac{e^{2x}\,\ln(5x)}{\log_{3}x}. Which of the following is an expression for p′(x)p'(x)? (a) (2e2xln⁡(5x)+e2xx)ln⁡xln⁡3+e2xln⁡(5x)xln⁡3(ln⁡xln⁡3)2\displaystyle \frac{\left(2e^{2x}\ln(5x)+\dfrac{e^{2x}}{x}\right)\dfrac{\ln x}{\ln 3}+\dfrac{e^{2x}\ln(5x)}{x\ln 3}}{\left(\dfrac{\ln x}{\ln 3}\right)^{2}} (b) (2e2xln⁡(5x)+e2xx)ln⁡xln⁡3−e2xln⁡(5x)x(ln⁡xln⁡3)2\displaystyle \frac{\left(2e^{2x}\ln(5x)+\dfrac{e^{2x}}{x}\right)\dfrac{\ln x}{\ln 3}-\dfrac{e^{2x}\ln(5x)}{x}}{\left(\dfrac{\ln x}{\ln 3}\right)^{2}} (c) (2e2xln⁡(5x)+e2xx)ln⁡xln⁡3−e2xln⁡(5x)xln⁡3(ln⁡xln⁡3)2\displaystyle \frac{\left(2e^{2x}\ln(5x)+\dfrac{e^{2x}}{x}\right)\dfrac{\ln x}{\ln 3}-\dfrac{e^{2x}\ln(5x)}{x\ln 3}}{\left(\dfrac{\ln x}{\ln 3}\right)^{2}} (d) (2e2xln⁡(5x)−e2xx)ln⁡xln⁡3−e2xln⁡(5x)xln⁡3(ln⁡xln⁡3)2\displaystyle \frac{\left(2e^{2x}\ln(5x)-\dfrac{e^{2x}}{x}\right)\dfrac{\ln x}{\ln 3}-\dfrac{e^{2x}\ln(5x)}{x\ln 3}}{\left(\dfrac{\ln x}{\ln 3}\right)^{2}}

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