Problem 29 - University Calculus: Early Transcendentals

Question

Find the derivatives of the function. $$y=2 e^{-x}+e^{3 x}$$

Step-by-Step Solution

Answer

The derivative is \(-2e^{-x} + 3e^{3x}\).

Step 1: Identify the Terms

The function is \( y = 2e^{-x} + e^{3x} \). We need to find the derivative of this function, which involves differentiating both terms separately: \( 2e^{-x} \) and \( e^{3x} \).

Step 2: Differentiate the First Term

Differentiate \( 2e^{-x} \) with respect to \( x \). The derivative of \( e^{-x} \) is \( -e^{-x} \), so the derivative of \( 2e^{-x} \) is \( -2e^{-x} \).

Step 3: Differentiate the Second Term

Now differentiate \( e^{3x} \) with respect to \( x \). Using the chain rule, the derivative of \( e^{u} \) is \( e^{u} \cdot u' \), and here \( u = 3x \), so \( u' = 3 \). Thus, the derivative is \( 3e^{3x} \).

Step 4: Combine the Derivatives

Add the derivatives from Steps 2 and 3 to get the derivative of the whole function. The derivative of \( y \) is \( \frac{dy}{dx} = -2e^{-x} + 3e^{3x} \).

Key Concepts

Chain Rule

The chain rule is a fundamental method in calculus used to differentiate compositions of functions. Here’s a way to think about it: if you have a function nested inside another function, the chain rule helps you find the derivative of the whole expression.

For example, if you have a function like \( e^{3x} \), this involves an "inner function", \( 3x \), and an "outer function", \( e^u \). The chain rule states that the derivative of the composition \( e^{3x} \) is given by the derivative of the outer function \( e^u \) evaluated at the inner function, multiplied by the derivative of the inner function \( u \).

Steps involved include:

• Identify the inner function, which is \( u = 3x \) in our case.
• Find the derivative of the inner function \( u' = 3 \).
• Apply the chain rule: derivative of \( e^{3x} \) is \( e^{3x} \times 3 = 3e^{3x} \).

Easy as that! The chain rule is powerful not only for exponential functions but for trigonometric and logarithmic functions as well.

Differentiation

Differentiation is the process of finding the derivative, which can be thought of as finding the rate of change of a function. In simple terms, given a function that describes a curve, differentiation helps us determine how steep that curve is at any point.

When you differentiate a function, you follow a set of rules depending on the types of terms present in the function. In our exercise, we differentiate each term of the function separately.

By working through the function \( y = 2e^{-x} + e^{3x} \), we differentiate the terms:

• \(2e^{-x}\): Here, the derivative is found based on the rule that states \( \frac{d}{dx}(e^u) = e^u u' \), where \( u = -x \) leading to \( u' = -1 \), so its derivative is \(-2e^{-x}\).
• \(e^{3x}\): Already discussed in the chain rule section, this gives \(3e^{3x}\).

This step-by-step differentiation approach helps you tackle any combination of functions confidently.

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable power. The function \(e^x\) is the most common exponential function, where \(e\) is an irrational constant approximately equal to 2.71828. Exponentials are fundamental in modeling growth processes, such as populations and investments.

When differentiating exponential functions, the derivative of \(e^u\) with respect to \(x\) is \(e^u \cdot u'\). This property is unique because the base \(e\) retains its form even after differentiation, unlike other bases.

Let's consider the function \(y = 2e^{-x} + e^{3x}\). Using properties of exponential differentiation:

• For \(2e^{-x}\), recognizing that any constant multiplier "c" of an exponential function remains: \(c \cdot (-1)e^{-x} = -2e^{-x}\).
• For \(e^{3x}\), as calculated before, the result is \(3e^{3x}\).

These rules make it straightforward to handle exponential terms, an essential skill since these functions appear frequently in fields like biology and finance.