Problem 4 - Applied Calculus

Question

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(f(x)=x^{3}+3^{x}\)

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 3x^2 + 3^x \cdot \ln(3) \).

Step 1: Differentiate the Power Function

Let's begin by differentiating the first term of the function, which is a power function: \( x^3 \). The derivative of \( x^n \) is \( nx^{n-1} \). Thus, the derivative of \( x^3 \) is \( 3x^{2} \).

Step 2: Differentiate the Exponential Function

Now, differentiate the second term \( 3^x \). The derivative of \( a^x \), where \( a eq e \), is given by \( a^x \cdot \ln(a) \). Therefore, the derivative of \( 3^x \) is \( 3^x \cdot \ln(3) \).

Step 3: Combine the Derivatives

To find the derivative \( f'(x) \), combine the derivatives of the two terms found in Steps 1 and 2. Thus, \( f'(x) = 3x^2 + 3^x \cdot \ln(3) \).

Key Concepts

Power Functions

Power functions are mathematical functions of the form \( f(x) = x^n \) where \( n \) is a constant real number. These are among the simplest types of functions you will encounter in calculus. Differentiating power functions is straightforward.

The derivative of a power function \( x^n \) is given by the formula \( nx^{n-1} \). This means you multiply the function by the exponent and then subtract 1 from the exponent. For example, for \( x^3 \), the derivative is \( 3x^2 \).

Some key points about power functions include:

• They have the form \( x^n \); \( n \) can be any real number, positive or negative.
• The power tells you how many times to multiply \( x \) by itself.
• The derivative rule allows you to calculate instant rates of change for these functions effortlessly.

Understanding power functions and their derivatives is foundational for solving more complex calculus problems.

Exponential Functions

Exponential functions have the general form \( a^x \), where \( a \) is a positive constant. These functions are not just limited to the number \( e \) but can be any constant greater than zero. Exponential functions grow at a rate proportional to their value, which makes them unique compared to polynomials.

The derivative of an exponential function \( a^x \) is given by \( a^x \cdot \ln(a) \). This tells us that when differentiating exponential functions, it involves both the original function and the natural logarithm of the base \( a \).

Important features of exponential functions:

• They have rapid rates of growth, often faster than polynomial functions.
• The derivative maintains the exponential form, ensuring similar behavior in their graphs.
• Exponential functions are common in real-world applications like population growth and radioactive decay.

Comprehending these functions empowers you to analyze situations involving continuous growth or decay accurately.

Chain Rule

The chain rule is a critical tool in calculus for differentiating compositions of functions. When you have a function inside another function, like \( f(g(x)) \), the chain rule is necessary. It helps us break down the differentiation process into manageable steps.

The chain rule states: if you have a composite function \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \). This means you differentiate the outer function while keeping the inner function unchanged, then multiply by the derivative of the inner function.

Points to remember about the chain rule:

• It is essential for differentiating composite functions.
• Allows you to tackle more complex derivatives that aren't straightforward.
• Frequently used across various calculus applications, enhancing analytical capabilities.

Mastering the chain rule is crucial for advancing in calculus and effectively solving diverse mathematical problems.