Question

Let f be a differentiable function satisfying the condition \[f\left( {\dfrac{x}{y}} \right) = \dfrac{{f\left( x \right)}}{{f\left( y \right)}}\] , for all \[x,y\left( { \ne 0} \right) \in R\] and \[f\left( y \right) \ne 0\] . If \[f'\left( 1 \right) = 2\] , then \[f'\left( x \right)\] is equal to
A \[2f\left( x \right)\]
B \[\dfrac{{f\left( x \right)}}{x}\]
C \[2xf\left( x \right)\]
D \[\dfrac{{2f\left( x \right)}}{x}\]

Answer 1

Hint: A function defines one variable in terms of another. A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable). Here, in the given statement, f is a function of both x and y, hence with respect to the given terms, we need to find \[f'\left( x \right)\] .

Complete step-by-step answer:
Given,
 \[f\left( {\dfrac{x}{y}} \right) = \dfrac{{f\left( x \right)}}{{f\left( y \right)}}\]
 \[f\left( x \right) = {x^m}\] only satisfies the given equation.
And
 \[f'\left( 1 \right) = 2\]
 \[ \Rightarrow \] \[m{\left( 1 \right)^{m - 1}} = 2\]
 \[ \Rightarrow \] \[m = 2\] \[ \Rightarrow \] \[f\left( x \right) = {x^2}\]
Hence, now differentiate the function \[f\left( x \right)\] with respect to x, we know that, the derivative of \[{x^2}\] is equal to 2x, hence we get:
 \[f'\left( x \right) = 2x\]
 \[f'\left( x \right) = 2\dfrac{{f\left( x \right)}}{x}\]
 \[ \Rightarrow \] \[f'\left( x \right) = \dfrac{{2f\left( x \right)}}{x}\]
Hence, option D is the right answer.

Note: A differential equation contains derivatives which are either partial derivatives or ordinary derivatives. The derivative represents a rate of change, and the differential equation describes a relationship between the quantity that is continuously varying with respect to the change in another quantity.
Note: While finding the derivative we must know all the basic differentiation formulas of the function, and here some of the formulas to note:
The derivative of \[{e^x}\] is equal to:
 \[\dfrac{{d\left( {{e^x}} \right)}}{{dx}} = {e^x}\]
The derivative of \[{x^n}\] is equal to:
 \[\dfrac{{d\left( {{x^n}} \right)}}{{dx}} = n{x^{n - 1}}\]
The derivative of \[{x^2}\] is equal to:
 \[\dfrac{{d\left( {{x^2}} \right)}}{{dx}} = 2x\]
The derivative of ax is equal to a:
 \[\dfrac{{d\left( {ax} \right)}}{{dx}} = a\] , in which a is a constant term.
The derivative represents a rate of change, and the differential equation describes a relationship between the quantity that is continuously varying with respect to the change in another quantity.