Question

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

Answer 1

Hint: 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: \[d\left( \dfrac{u}{v} \right)=\dfrac{v\cdot du-u\cdot dv}{{{v}^{2}}}\]
\[d\left( \ln x \right)=\dfrac{1}{x}dx\]