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(d)/(dx)(log_(10)x+log_(x)10+log_(x)x+log_(10)10)

$11$ ) $\frac{d}{d x}\left(\log _{10} x+\log _{x} 10+\log _{x} x+\log _{10} 10\right)$

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Quotient property of logarithms

Full solution

Q.

11

)

\frac{d}{d x}\left(\log _{10} x+\log _{x} 10+\log _{x} x+\log _{10} 10\right)

Find Derivative of $\log_{10}x$ : We need to find the derivative of each term separately. $\newline$ First, let's find the derivative of $\log_{10}x$ . $\newline$ Using the change of base formula, $\log_{10}x = \frac{\ln(x)}{\ln(10)}$ . $\newline$ The derivative of $\ln(x)$ is $\frac{1}{x}$ , so the derivative of $\log_{10}x$ is $\frac{1}{x\ln(10)}$ .
Find Derivative of $\log_{x}10$ : Now, let's find the derivative of $\log_{x}10$ . This is an inverse logarithm, so we use the formula $\frac{d}{dx}(\log_{x}a) = -\frac{1}{x\ln(a)}$ . Here, $a = 10$ , so the derivative of $\log_{x}10$ is $-\frac{1}{x\ln(10)}$ .
Find Derivative of $\log_{x}x$ : Next, we find the derivative of $\log_{x}x$ . This is a logarithm with the base the same as the argument, which simplifies to $1$ . The derivative of a constant is $0$ .
Find Derivative of $\log_{10}10$ : Lastly, we find the derivative of $\log_{10}10$ . This is a constant because it's the log of a number with the same base, so its derivative is also $0$ .
Sum of Derivatives: Adding up all the derivatives, we get $\frac{1}{x\ln(10)} - \frac{1}{x\ln(10)} + 0 + 0$ . This simplifies to $0$ , since the first two terms cancel each other out.

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