Problem 36 - Calculus and its applications

Question

Differentiate. $$ f(x)=\left(x^{2}+3 x-9\right) e^{x} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = (x^2 + 5x - 6)e^x. \)

Step 1: Identify the Differentiation Rule Needed

We notice that the function is a product of two functions: \( g(x) = x^2 + 3x - 9 \) and \( h(x) = e^x \). Hence, we need to use the product rule for differentiation, which is \( (u imes v)' = u' imes v + u imes v' \).

Step 2: Differentiate the First Part of the Product

Identify \( g(x) = x^2 + 3x - 9 \) and differentiate it. \[ \frac{d}{dx}(g(x)) = \frac{d}{dx}(x^2 + 3x - 9) = 2x + 3. \] This is \( g'(x) \).

Step 3: Differentiate the Second Part of the Product

Identify \( h(x) = e^x \) and find its derivative. \[ \frac{d}{dx}(h(x)) = \frac{d}{dx}(e^x) = e^x. \] This is \( h'(x) \).

Step 4: Apply the Product Rule

Substitute \( g(x), g'(x), h(x), \) and \( h'(x) \) into the product rule formula: \[ (x^2 + 3x - 9)' imes e^x + (x^2 + 3x - 9) imes (e^x)' = (2x + 3) e^x + (x^2 + 3x - 9) e^x. \]

Step 5: Simplify the Expression

Factor out \( e^x \) from the terms: \[ f'(x) = e^x (2x + 3 + x^2 + 3x - 9). \] Simplify inside the parentheses to get: \[ e^x (x^2 + 5x - 6). \] Therefore, the derivative of the function is \( f'(x) = (x^2 + 5x - 6)e^x. \)

Key Concepts

Product Rule

When faced with differentiating a product of two functions, the product rule comes to the rescue. This technique is crucial for handling products in calculus. Suppose you have two differentiable functions, represented as \( u(x) \) and \( v(x) \). The product rule then states:\[(u \times v)' = u' \times v + u \times v'\]Here's what each term means:

• \( u' \): the derivative of the first function.
• \( v' \): the derivative of the second function.
• \( u \times v' \): multiply the original first function by the derivative of the second.
• \( u' \times v \): multiply the derivative of the first by the original second function.

By using the product rule, you can differentiate a product of functions accurately by accounting for changes in both parts of the product. This process ensures that you capture all aspects of how the functions interact as they change.

Derivatives

Derivatives fundamentally represent how a function is changing at any point, forming the backbone of calculus. They measure the rate at which a function's value changes as its input changes. The derivative of a function \( f(x) \) is commonly denoted as \( f'(x) \) or \( \frac{df}{dx} \).Understanding derivatives involves grasping these key concepts:

• Power Rule: The derivative of \( x^n \) is \( nx^{n-1} \). This rule simplifies finding derivatives of polynomial functions.
• Constant Rule: The derivative of a constant is zero since constants do not change.
• Sum Rule: The derivative of a sum is the sum of the derivatives: \( (f + g)' = f' + g' \).

By applying these basic rules and the product rule, you can build the derivative of more complex functions step by step. Derivatives not only help identify slopes at various points but also set the foundation for advanced topics like optimization and curve sketching.

Exponential Function

The exponential function, denoted as \( e^x \), is a special type of function where the base of the exponential is the constant \( e \), approximately equal to 2.71828. What makes \( e^x \) particularly remarkable is that its derivative is the same as the original function:\[ \frac{d}{dx}(e^x) = e^x \]This property makes the exponential function tremendously useful across many fields. Here’s what you need to remember:

• Growth Rate: The function \( e^x \) grows quickly as \( x \) increases.
• Coefficient Rule: If you have \( ce^x \) where \( c \) is a constant, the derivative is simply \( ce^x \).
• Chain Rule: For \( e^{f(x)} \), the derivative is \( e^{f(x)} \cdot f'(x) \), using the chain rule.

Exponential functions appear in contexts ranging from population models to finance due to their nature of continuous growth. Being familiar with their properties and derivatives enhances your ability to tackle problems involving natural growth and decay.