Question

If \[f(x)\] and \[g(x)\] are two function with \[g(x)=x-\dfrac{1}{x}\] and \[fog(x)={{x}^{3}}-\dfrac{1}{{{x}^{3}}}\] , then \[{{f}^{'}}(x)\] is equal to
A. \[3{{x}^{2}}+3\]
B. \[{{x}^{2}}-\dfrac{1}{x}\]
C. \[1-\dfrac{1}{{{x}^{2}}}\]
D. \[3{{x}^{2}}-\dfrac{3}{{{x}^{4}}}\]

Answer 1

Hint: In this particular problem there are two function are given that is \[g(x)=x-\dfrac{1}{x}\] and \[fog(x)={{x}^{3}}-\dfrac{1}{{{x}^{3}}}\] in that we have find \[f(x)\] and differentiate to find the value of \[{{f}^{'}}(x)\] .
So, in this way we have to solve the problems.

Complete step-by-step answer:
In this type of problems they have mentioned the two function that is \[g(x)=x-\dfrac{1}{x}\] and \[fog(x)={{x}^{3}}-\dfrac{1}{{{x}^{3}}}--(1)\]
S, we have to find the \[{{f}^{'}}(x)\]
 For that it is necessary to find the \[f(x)\]
First of all we need to modify the equation (1) that is \[fog(x)={{x}^{3}}-\dfrac{1}{{{x}^{3}}}--(1)\]
To modify this we need to add or subtract \[3x\] and also we need to add or subtract \[\dfrac{3}{x}\] in above equation (1)
 \[fog(x)={{x}^{3}}-\dfrac{1}{{{x}^{3}}}-3x+3x+\dfrac{3}{x}-\dfrac{3}{x}\]
By rearranging the term on the equation (1)
 \[fog(x)={{x}^{3}}-\dfrac{1}{{{x}^{3}}}-3x+\dfrac{3}{x}+3x-\dfrac{3}{x}\]
To get appropriate function above equation can also be written in the form of
 \[fog(x)={{x}^{3}}-\dfrac{1}{{{x}^{3}}}-3\left( {{x}^{2}}\times \dfrac{1}{x} \right)+3\left( x\times \dfrac{1}{{{x}^{2}}} \right)+3x-\dfrac{3}{x}\]
If you notice this above equation then it is in the form of \[{{(a-b)}^{3}}={{a}^{3}}-3{{a}^{2}}b+3a{{b}^{2}}-{{b}^{3}}\] . So, we apply this property in above equation we get:
 \[fog(x)={{\left( x-\dfrac{1}{x} \right)}^{3}}+3x-\dfrac{3}{x}\]
By modifying this equation we get:
 \[fog(x)={{\left( x-\dfrac{1}{x} \right)}^{3}}+3\left( x-\dfrac{1}{x} \right)\]
If you notice the above equation then the above function is in the term of \[\left( x-\dfrac{1}{x} \right)\] but we want the term in x only we need to find \[f(x)\] that is
 \[f(x)={{x}^{3}}+3x\]
Now, we have to differentiate with respect to x
We get:
 \[{{f}^{'}}(x)=3{{x}^{2}}+3\]
So, the correct option is “option A”.
So, the correct answer is “Option A”.

Note: In this type of problem we need to keep in mind that first of all we have to find \[f(x)\] to find the value of \[{{f}^{'}}(x)\] . We have to expand \[fog(x)\] by using the basic property and get the desired answer. Here, \[fog(x)\] is a composite-function and \[fog(x)\] means \[g(x)\] function is in \[f(x)\] function. To find the first derivative then we have to differentiate \[f(x)\] which we get from expanding the \[fog(x)\] .