Problem 246 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } f(x)=e^{x} g(x), g(0)=2, g^{\prime}(0)=1, \text { then find } f^{\prime}(0) \text { . } $$

Step-by-Step Solution

Answer

The derivative of \(f(x) = e^x g(x)\) at \(x = 0\) is \(f'(0) = 3\).

Step 1: Apply the Product Rule for Differentiation

Knowing that \(f(x) = e^x * g(x)\), we'll apply the product rule for differentiation: \( (h * k)' = h' * k + h * k' \). Here, \(h(x) = e^x\) and \(k(x) = g(x)\).

Step 2: Differentiate h(x) with respect to x

Differentiate \(h(x) = e^x\) with respect to x: \[ h'(x) = \frac{d}{dx}(e^x) = e^x. \]

Step 3: Incorporate the given information about g(x)

The problem has given us information about the function \(g(x)\) that we will need for our final calculation: \[ g(0) = 2, \quad g'(0) = 1. \]

Step 4: Find f'(x) using the product rule

Using the product rule, the derivative of \(f(x)\) is: \[ f'(x) = h'(x)k(x) + h(x)k'(x) = e^x g(x) + e^x g'(x). \]

Step 5: Find f'(0)

Calculate the value of the derivative of f(x) at \(x = 0\): \[ f'(0) = e^0 g(0) + e^0 g'(0) = (1)(2) + (1)(1) = 3. \] Thus, the derivative of \(f(x) = e^x g(x)\) at \(x = 0\) is \(f'(0) = 3\).

Key Concepts

Product Rule

The product rule is a crucial concept in calculus used to differentiate expressions where two functions are multiplied together. When you have a function that is the product of two other functions, say \( h(x) \) and \( k(x) \), the product rule helps us find the derivative of the product. In mathematical terms, if \( f(x) = h(x)k(x) \), then the derivative of \( f(x) \) is given by:

• \( f'(x) = h'(x)k(x) + h(x)k'(x) \).

In this formula:

• \( h'(x) \) is the derivative of \( h(x) \),
• \( k'(x) \) is the derivative of \( k(x) \),
• and \( h'(x)k(x) + h(x)k'(x) \) combines these derivatives wisely.

Break up the task by handling each function separately. This systematically applies derivatives to both components of the product, ensuring no part is overlooked. Remember, practice applying the product rule with various functions to become comfortable with the differentiation process.

Exponential Function

An exponential function is one of the most common and significant types of functions in mathematics. In its simplest form, an exponential function is a function of the type \( f(x) = a^x \), where \( a \) is a positive constant. The most famous and commonly used base is \( e \), where the function becomes \( f(x) = e^x \). The exponential function \( f(x) = e^x \) has some unique properties:

• Its derivative is \( e^x \), meaning the rate of change of the function is proportional to its value.
• \( e^x \) grows or decays very rapidly, which makes it essential in modeling real-world phenomena like population growth or radioactive decay.
• It's always positive, reflecting its continuous growth.

This characteristic of retaining its form upon differentiation makes \( e^x \) particularly special. For example, when differentiating \( e^x \) as part of another function, such as in a product form, this unique trait remains, simplifying calculations.

Function Derivative

The derivative of a function is a fundamental concept in calculus that measures how a function changes as its input changes. In other words, it gives the rate of change or the slope of the function at any given point. Calculating this derivative helps to understand the behavior of functions across their domain.

• For any function \( f(x) \), the derivative is denoted as \( f'(x) \).
• The process of finding a derivative is called differentiation.

Differentiation has rules, like the product rule, that allow us to deal with more complex functions:

• Simple rules for derivatives help differentiate basic functions such as polynomials, trigonometric, and exponential functions.
• Advanced rules, such as the product, quotient, or chain rules, aid in differentiating more complex expressions.

By fastening all these principles together, finding derivatives becomes a strategic step-by-step method where each rule comes into play when needed, leading to a deeper comprehension of changes within any given functions.