Question

Find the derivative of $f\left( x \right)=\dfrac{7\ln x}{4x}$?

Answer 1

This question is from the topic of differentiation. In this question, we have to find the derivative of $f\left( x \right)$ that is $f'\left( x \right)$ or $\dfrac{d}{dx}\left[ f\left( x \right) \right]$. In solving this question, we will first differentiate the equation $f\left( x \right)=\dfrac{7\ln x}{4x}$ using the formula of division rule of differentiation.

Complete step by step solution:
Let us solve this question.
In this question, we have asked to find the differentiation of $f\left( x \right)=\dfrac{7\ln x}{4x}$.
So, the differentiation of $f\left( x \right)=\dfrac{7\ln x}{4x}$ will be like
$d\left[ f\left( x \right) \right]=d\left( \dfrac{7\ln x}{4x} \right)$
As we can see that there is a constant in numerator and denominator in the right side of equation, so we can write the above as
\Rightarrow d\left[ f\left( x \right) \right]=\dfrac{7}{4}\left\\{ d\left( \dfrac{\ln x}{x} \right) \right\\}
Now, this can be solved by using the division rule of differentiation. The formula for division rule of differentiation is $d\left( \dfrac{u}{v} \right)=\dfrac{v\cdot du-u\cdot dv}{{{v}^{2}}}$.
So, we can write
\Rightarrow d\left[ f\left( x \right) \right]=\dfrac{7}{4}\left\\{ \dfrac{x\cdot d\left( \ln x \right)-\left( \ln x \right)\cdot dx}{{{x}^{2}}} \right\\}
Using the formula of differentiation that is $d\left( \ln x \right)=\dfrac{1}{x}dx$, we can write the above equation as
\Rightarrow d\left[ f\left( x \right) \right]=\dfrac{7}{4}\left\\{ \dfrac{x\cdot \dfrac{1}{x}dx-\left( \ln x \right)\cdot dx}{{{x}^{2}}} \right\\}
The above equation can also be written as
\Rightarrow d\left[ f\left( x \right) \right]=\dfrac{7}{4}\left\\{ \dfrac{dx-\left( \ln x \right)\cdot dx}{{{x}^{2}}} \right\\}
Now, taking ‘dx’ as common to the both side of the equation, we can write
\Rightarrow d\left[ f\left( x \right) \right]=\dfrac{7}{4}\left\\{ \dfrac{1-\ln x}{{{x}^{2}}} \right\\}dx
Now, dividing ‘dx’ to the both side of the equation, we get
\Rightarrow \dfrac{d}{dx}\left[ f\left( x \right) \right]=\dfrac{7}{4}\left\\{ \dfrac{1-\ln x}{{{x}^{2}}} \right\\}
The above equation can also be written as
\Rightarrow \dfrac{d}{dx}\left[ f\left( x \right) \right]=\dfrac{7}{4}\left\\{ \dfrac{1}{{{x}^{2}}}-\dfrac{\ln x}{{{x}^{2}}} \right\\}
The above equation can also be written as
$\Rightarrow \dfrac{d}{dx}\left[ f\left( x \right) \right]=\dfrac{7}{4}\dfrac{1}{{{x}^{2}}}-\dfrac{7}{4}\dfrac{\ln x}{{{x}^{2}}}$
$\Rightarrow \dfrac{d}{dx}\left[ f\left( x \right) \right]=\dfrac{7}{4{{x}^{2}}}-\dfrac{7\ln x}{4{{x}^{2}}}$
We can write $\dfrac{d}{dx}\left[ f\left( x \right) \right]$ as $f'\left( x \right)$, so we can write the above equation as
$\Rightarrow f'\left( x \right)=\dfrac{7}{4{{x}^{2}}}-\dfrac{7\ln x}{4{{x}^{2}}}$
Hence, we have found the derivative of $f\left( x \right)=\dfrac{7\ln x}{4x}$. The derivative of $f\left( x \right)=\dfrac{7\ln x}{4x}$ is $f'\left( x \right)=\dfrac{7}{4{{x}^{2}}}-\dfrac{7\ln x}{4{{x}^{2}}}$.

Note: As we can see that this question is from the topic of differentiation, so we should have a better knowledge in that topic. Always remember that whenever there is a constant multiplied in the differentiation, then constant terms will be taken out from the differentiation and then can do the further differentiation. Let us understand this from the following example:
$d\left( n\cdot x \right)=n\cdot dx$, where n is a constant. Here, we can see that we have taken out the constant term.
Remember the following formulas:
Product rule of differentiation: $d\left( \dfrac{u}{v} \right)=\dfrac{v\cdot du-u\cdot dv}{{{v}^{2}}}$
$d\left( \ln x \right)=\dfrac{1}{x}dx$