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# How do you find the derivative of $\ln \left( {\dfrac{3}{x}} \right)$ with respect to x ?

Last updated date: 12th Aug 2024
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Hint: In the given problem, we are required to differentiate $\ln \left( {\dfrac{3}{x}} \right)$ with respect to x. Since, $\ln \left( {\dfrac{3}{x}} \right)$ is a composite function, so we will have to apply chain rule of differentiation in the process of differentiating $\ln \left( {\dfrac{3}{x}} \right)$ . So, differentiation of $\ln \left( {\dfrac{3}{x}} \right)$ with respect to x will be done layer by layer using the chain rule of differentiation. Also the derivative of $\ln \left( x \right)$with respect to $x$ must be remembered.

To find derivative of $\ln \left( {\dfrac{3}{x}} \right)$ with respect to x we have to find differentiate $\ln \left( {\dfrac{3}{x}} \right)$with respect to x.
So, Derivative of $\ln \left( {\dfrac{3}{x}} \right)$ with respect to x can be calculated as $\dfrac{d}{{dx}}\left( {\ln \left( {\dfrac{3}{x}} \right)} \right)$ .
Now, $\dfrac{d}{{dx}}\left( {\ln \left( {\dfrac{3}{x}} \right)} \right)$
First we differentiate $\ln \left( {\dfrac{3}{x}} \right)$ with respect to $\left( {\dfrac{3}{x}} \right)$, and then differentiate $\left( {\dfrac{3}{x}} \right)$ with respect to x.
Now, Let us assume $u = \left( {\dfrac{3}{x}} \right)$. So substituting $\left( {\dfrac{3}{x}} \right)$as $u$, we get,
$=$$\dfrac{d}{{dx}}\left( {\ln \left( u \right)} \right)$
Now, we know that the derivative of $\ln t$ with respect to $t$ is $\left( {\dfrac{1}{t}} \right)$. Hence, we get,
$= $$\dfrac{1}{u}\dfrac{{du}}{{dx}} Now, putting back uas \left( {\dfrac{3}{x}} \right), we get, =$$\dfrac{1}{{\left( {\dfrac{3}{x}} \right)}}\dfrac{{d\left( {\dfrac{3}{x}} \right)}}{{dx}}$ because $\dfrac{{du}}{{dx}} = \dfrac{{d\left( {\dfrac{3}{x}} \right)}}{{dx}}$
Now, we know the power rule of differentiation. Hence, we can apply the power rule so as to calculate the derivative of $\left( {\dfrac{3}{x}} \right)$ with respect to x.
$= $$\dfrac{x}{3}\left( {\dfrac{{ - 3}}{{{x^2}}}} \right) Cancelling the term in numerator and denominator, we get, =$$\left( {\dfrac{{ - 1}}{x}} \right)$
So, the derivative of $\ln \left( {\dfrac{3}{x}} \right)$ with respect to $x$is $\left( {\dfrac{{ - 1}}{x}} \right)$.