Problem 120 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(g\) is the inverse of a function \(f\) and \(f^{\prime}(x)=\frac{1}{1+x^{5}}\), then \(g^{\prime}(x)\) is equal to (A) \(1+x^{5}\) (B) \(5 x^{4}\) (C) \(\frac{1}{1+\\{g(x)\\}^{5}}\) (D) \(1+\\{g(x)\\}^{5}\)

Step-by-Step Solution

Answer

The answer is (D) \( 1 + g(x)^{5} \).

Step 1: Understand the Relationship between Inverse Functions and Derivatives

When g is the inverse of f, then for any x, if y = g(x) then x = f(y). We know the relationship between the derivatives of inverse functions: if g is the inverse of f, then \( g^{\prime}(x) = \frac{1}{f^{\prime}(g(x))} \).

Step 2: Identify Given Derivative

We are given that \( f^{\prime}(x) = \frac{1}{1+x^{5}} \). Our task is to find \( g^{\prime}(x) \) using the relationship identified earlier.

Step 3: Apply the Derivative of Inverse Relationship

According to the formula, \( g^{\prime}(x) = \frac{1}{f^{\prime}(g(x))} \). Substitute the expression for \( f^{\prime}(x) \), so \( g^{\prime}(x) = \frac{1}{\frac{1}{1+g(x)^{5}}} \).

Step 4: Simplify the Expression

The equation simplifies to \( g^{\prime}(x) = 1 + g(x)^{5} \) since \( \frac{1}{\frac{1}{1+g(x)^{5}}} = 1 + g(x)^{5} \).

Step 5: Choose the Correct Option

Refer back to the options given. This expression, \( g^{\prime}(x) = 1 + g(x)^{5} \), corresponds to option D.

Key Concepts

Derivatives of Inverse Functions

Understanding the derivatives of inverse functions might seem tricky at first, but it's quite fascinating! Here’s the scoop: if you have a function, say \( f \), with an inverse function \( g \), the derivatives of these functions are connected in a unique way. This relationship helps us understand how changes in one function reflect as changes in its inverse.

Here's the magic formula for their derivatives: if \( g \) is the inverse of \( f \), then \( g^{\prime}(x) = \frac{1}{f^{\prime}(g(x))} \).

Let's visualize this as a see-saw. If one side changes, the other side adjusts in a specific reciprocal manner. This derivative relationship is essential in calculus as it allows us to easily find the derivative of an inverse function given the derivative of the original function.

Function Derivatives

Function derivatives are like the speedometers of mathematics—they tell us how fast or slow a function is changing at any given point in its domain. In a more technical sense, the derivative of a function at any point gives us the slope of the tangent line to the graph of the function at that point.

For instance, for a function \( f(x) \), its derivative \( f^{\prime}(x) \) is calculated using the limit process:

• First, consider a small change in \( x\), then observe the change in \( f(x) \).
• Use the limit \( \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \) to find the derivative.

This essence of function derivatives serves as the building block for understanding more complex concepts in calculus, such as the derivatives of inverse functions.

Inverse Function Derivative Formula

The derivative formula for inverse functions is a powerful tool. When you need to find the derivative of an inverse function, this formula makes the task much simpler. Let's break it down: Suppose \( g \) is the inverse of a function \( f \) and we need \( g^{\prime}(x) \). You can use the derivative of \( f \) to find it with this formula: \( g^{\prime}(x) = \frac{1}{f^{\prime}(g(x))} \).

In our specific scenario, we're given \( f^{\prime}(x) = \frac{1}{1+x^{5}} \). Plugging this into the inverse function derivative formula leads us to:

• Substitute \( g(x)\), resulting in \( g^{\prime}(x) = \frac{1}{\frac{1}{1+g(x)^{5}}} \).
• Simplify this expression to get \( g^{\prime}(x) = 1 + g(x)^{5} \).

The simplicity of this formula is its beauty, as it allows the complex process of differentiation to become manageable.