Problem 37 - A Complete Resource Book in Mathematics for JEE Main

Question

A function \(f(x)\) is so defined that for all \(x,[f(x)]^{n}=\) \(f(n x)\). If \(f^{\prime}(x)\) denotes derivative of \(f(x)\) with respect to \(x\), then \(f^{\prime}(x) \times f(n x)=\) (A) \(f(x)\) (B) 0 (C) \(f(x) \times f^{\prime}(n x)\) (D) None of these

Step-by-Step Solution

Answer

The answer is (C) \(f(x) \times f^{\prime}(n x)\).

Step 1: Differentiate the given function implicitly

Start with the function equation given: \( [f(x)]^n = f(nx) \). Differentiate both sides with respect to \(x\) using implicit differentiation: \(\frac{d}{dx}[f(x)]^n = \frac{d}{dx}f(nx)\)

Step 2: Apply the chain rule for differentiation

For \(\frac{d}{dx}[f(x)]^n\), use the chain rule: \(n[f(x)]^{n-1}f'(x)\). For the right side \(\frac{d}{dx}f(nx)\), also use the chain rule: \(f'(nx) \cdot n\) because \(d/nx = n\cdot dx\). Thus, we have:\[n[f(x)]^{n-1}f'(x) = n f'(nx)\].

Step 3: Simplify the derived equation

Factor out \(n\) from both sides and cancel it:\[[f(x)]^{n-1}f'(x) = f'(nx)\].Since \([f(x)]^{n-1}\) is a common term in our differential result, rearrange it:

Step 4: Analyze the implication given the functional form

Re-examine the equation \([f(x)]^{n-1}f'(x) = f'(nx)\). Multiply both sides by \(f(nx)\) to find the required expression:\[f'(x)\times f(nx) = [f(x)]^{n-1} \times f'(nx)\times f(nx) = f(x)f'(nx)\].

Step 5: Compare with given options

We derived that \(f'(x) \times f(nx) = f(x) \times f'(nx)\), which corresponds to option (C).

Key Concepts

Chain Rule

The Chain Rule is a fundamental tool in calculus used for differentiating composite functions. When you have a function nested inside another, like \( h(x) = g(f(x)) \), the Chain Rule helps find its derivative. For the function \( [f(x)]^n = f(nx) \), applying the Chain Rule entails accounting for how both the outer function and the inner function change.

In our example:

• The outer function is \((f(x))^n\), needing a derivative concerning \(f(x)\), which is \(n(f(x))^{n-1}\).
• The inner function is \(f(x)\), differentiating it with respect to \(x\) gives us \(f'(x)\).

Using the Chain Rule, you multiply the derivative of the outer function by the derivative of the inner function. That forms \(n(f(x))^{n-1}f'(x)\) for the left side.

Similarly, when differentiating \(f(nx)\), treat \(nx\) as the inner function. Hence,

• You find the derivative of \(f\), as \(f'(nx)\), then multiply it by the derivative of \(nx\) with respect to \(x\), which is \(n\).

This results in \(n(f'(nx))\) on the right side. The Chain Rule is crucial for correctly deriving the expression.

Function Derivation

When we talk about Function Derivation in calculus, it refers to finding the derivative of a function. This process reveals the rate at which the function's value changes with respect to its input.

In our scenario, we need to derive both sides of the equation \((f(x))^n = f(nx)\) to understand their rates of change. The derivative, noted as \(f'(x)\), is simply the function's slope at any given point. By establishing a connection between the function and its variables, we learn about its behaviors and trends.

Here's a step-by-step of how we differentiate this equation implicitly:

• On the left, differentiating \((f(x))^n\) results in \(n(f(x))^{n-1}f'(x)\) through the Chain Rule.
• The right side, \(f(nx)\), differentiates to \(f'(nx)\times n\).
• This shows both how the powers of \(f(x)\) reduce and how the derivative \(f'(x)\) emerges due to the underlying relationship that the equation defines.

Function Derivation not only assists in differentiating single components effectively but helps in ensuring comprehensive analysis of entire functional expressions.

Differential Equations

Differential Equations involve equations with derivatives in them. They express relationships between functions and their rates of change. Solving them often requires finding a function (or functions) that satisfies the given equation.

In the exercise, once we differentiate the expression \([f(x)]^n = f(nx)\), we have new equations that describe the relationship of the derivatives. These equations hold true for all values of \(x\):

• The derived equation becomes \([f(x)]^{n-1}f'(x) = f'(nx)\).
• This shows a special type of relationship between \(f\), \(f'\), and their functional transformations.
• Simplifying these differential equations assist us in finding solutions or answers to posed function-related problems, such as choosing the correct option from (A) to (D) in our case.

Differential Equations like the one we have analyzed are instrumental for advanced problems in physics, engineering, and many scientific fields where dynamic processes are mathematically modeled.