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The second order derivative of which of the following functions is 5x5^x5x​?
Question

The second order derivative of which of the following functions is 5x5^x​?

A.

5x(loge5)5^x(\log_e 5)​​

B.

5x(loge5)25^x(\log_e 5)^2​​

C.

5x(loge5)\frac{5^x}{(\log_e 5)}​​

D.

5x(loge5)2\frac{5^x}{(\log_e 5)^2}​​

Correct option is D

ddx(5x(loge5)2)=5xlog5(loge5)2=5xlog5Again, differentiatingd2dx2(5xlog5)=5xlog5log5=5x\begin{aligned}&\frac{d}{dx} \left( \frac{5^x}{(\log_e 5)^2} \right) = \frac{5^x \log 5}{(\log_e 5)^2} = \frac{5^x}{\log 5} \\&\text{Again, differentiating} \\&\frac{d^2}{dx^2} \left( \frac{5^x}{\log 5} \right) = \frac{5^x \log 5}{\log 5} = 5^x\end{aligned}​​

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