Problem 5 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

If \(y=e^{\left(1+\log _{c} x\right)}\), then \(\frac{d y}{d x}=\) [MPPET-1996; Pb.CET-2001] (a) \(e\) (b) 1 (c) 0 (d) \(\log _{e} x e^{\log \alpha x}\)

Step-by-Step Solution

Answer

The derivative does not match the given options after simplification.

Step 1: Understanding the Function

We start by identifying the function given: \(y = e^{(1 + \log_c x)}\). This is an exponential function where the exponent is \(1 + \log_c x\). Our goal is to find the derivative of this function with respect to \(x\).

Step 2: Transform the Logarithm Base from c to e

Convert the logarithm with base \(c\) to natural logarithms (base \(e\)) as we normally calculate derivatives using natural logarithms. The change of base formula is \(\log_c x = \frac{\log_e x}{\log_e c}\). So, the function becomes: \(y = e^{(1 + \frac{\log_e x}{\log_e c})}\).

Step 3: Apply the Chain Rule

To differentiate \(y = e^{(1 + \frac{\log_e x}{\log_e c})}\) with respect to \(x\), we use the chain rule. We'll treat \(1 + \frac{\log_e x}{\log_e c}\) as the inner function \(u\), i.e., \(u = 1 + \frac{\log_e x}{\log_e c}\).

Step 4: Differentiate the Inner Function

Find the derivative of the inner function \(u = 1 + \frac{\log_e x}{\log_e c}\). The derivative of \(1\) is \(0\) and the derivative of \(\frac{\log_e x}{\log_e c}\) with respect to \(x\) is \(\frac{1}{x \cdot \log_e c}\). Thus, \(\frac{du}{dx} = \frac{1}{x \cdot \log_e c}\).

Step 5: Differentiate the Exponential Function

The derivative of \(e^u\) with respect to \(x\) is given by \(\frac{dy}{dx} = e^u \cdot \frac{du}{dx}\). We already have \(u\) and its derivative, so plug in these values: \(\frac{dy}{dx} = e^{(1 + \frac{\log_e x}{\log_e c})} \cdot \frac{1}{x \cdot \log_e c}\).

Step 6: Simplify the Derivative

Simplify \(\frac{dy}{dx} = \frac{1}{x \cdot \log_e c} \cdot e^{(1 + \frac{\log_e x}{\log_e c})}\) to see if it matches any of the given options. On simplification, it does not match the options directly but it involves the exponential and logarithmic parts. Thus, no simplification matches the options.

Key Concepts

Chain Rule in Calculus

The chain rule is a fundamental concept in calculus, which allows us to differentiate complex functions. It is particularly useful when dealing with composite functions, where one function is nested inside another.
Here’s a straightforward way to think about it: imagine a function, let's call it \( y = f(g(x)) \). The chain rule helps in finding the derivative of \( y \) with respect to \( x \). To apply the rule, we differentiate the outer function, \( f \), with respect to the inner function, \( g(x) \), then multiply it by the derivative of the inner function, \( g(x) \), with respect to \( x \).
The formula for using the chain rule is:

• \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \)

In the original exercise, the chain rule is applied to differentiate \( y = e^{(1 + \frac{\log_e x}{\log_e c})} \). Here, the inner function is \( u = 1 + \frac{\log_e x}{\log_e c} \), and the outer function is \( e^u \). By differentiating each part separately, we can find the overall derivative.

Understanding Exponential Functions

Exponential functions are a vital part of calculus and are characterized by constant base numbers raised to varying powers, typically represented as \( e^x \), where \( e \) is a mathematical constant approximately equal to 2.718.
These functions grow at rates proportional to their current values, which makes them ideal for modeling growth and decay processes.

• The general form of an exponential function is \( y = a \cdot e^{bx} \), where the base \( a \) is a constant and \( b \) influences the growth rate.
• Differentiating exponential functions like \( e^x \) is straightforward, as the derivative of \( e^x \) with respect to \( x \) is \( e^x \).
In the problem at hand, we work with an exponential function with a composite exponent, \( e^{(1 + \frac{\log_e x}{\log_e c})} \). The exponential function plays a critical role when applying the chain rule to find the derivative, illustrating the powerful tools calculus offers to handle complex expressions.

Basics of Logarithmic Functions

Logarithms are the inverse of exponential functions, and they answer the question: "What exponent do we need to raise a base to, in order to get a certain number?"
The most commonly used logarithms are natural logarithms, base \( e \), denoted as \( \log_e(x) \) or simply \( \ln(x) \).

• The natural logarithm \( \ln(x) \) is used extensively in calculus because it simplifies the process of differentiation and integration.
• A crucial property of logarithms is the change of base formula: \( \log_c(x) = \frac{\log_e(x)}{\log_e(c)} \), which allows us to convert any logarithm to a natural logarithm.
• When differentiating \( \ln(x) \), the result is \( \frac{1}{x} \).

In the original exercise, the base \( c \) logarithm is converted to a natural logarithm using the change of base formula, which makes it easier to apply the chain rule and find the derivative.