Problem 337 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=2^{4 x}+4 x^{2}$$

Step-by-Step Solution

Answer

\( f'(x) = 4 \times 2^{4x} \ln(2) + 8x \)

Step 1: Differentiate the Exponential Term

To differentiate the function, start with the exponential term. Remember the derivative of an exponential function, which is \( a^{g(x)} \) where the derivative is \( a^{g(x)} \ln(a) \, g'(x) \). In this case, the term is \( 2^{4x} \), and you differentiate it as \( (2^{4x})' = 2^{4x} \ln(2) \times 4 \). Thus, the derivative is \( 4 \times 2^{4x} \ln(2) \).

Step 2: Differentiate the Polynomial Term

Next, differentiate the polynomial term \( 4x^2 \). Recall that the power rule \( \frac{d}{dx} x^n = nx^{n-1} \) can be applied here.\( \frac{d}{dx} 4x^2 = 2 \times 4x^{2-1} = 8x \).

Step 3: Combine the Results

Now, add the derivatives obtained in the previous steps. The derivative of the function \( f(x) \) is obtained by combining both the results: \( f'(x) = 4 \times 2^{4x} \ln(2) + 8x \).

Key Concepts

Exponential Differentiation

Exponential differentiation involves finding the derivative of functions with exponential expressions. When dealing with an exponential function like \( f(x) = a^{g(x)} \), the key is applying the formula for the derivative:

• Take the original exponential expression \( a^{g(x)} \).
• Multiply by the natural logarithm of the base \( \ln(a) \).
• Include the derivative of the exponent \( g'(x) \).

For example, in the function \( f(x) = 2^{4x} \), the base is 2 and the exponent is \( 4x \). Therefore, differentiate it as \( (2^{4x})' = 2^{4x} \ln(2) \cdot 4 \). This means we maintain the original form multiplied by both \( \ln(2) \) and the derivative of the exponent \( 4 \), resulting in \( 4 \times 2^{4x} \ln(2) \).
Understanding exponential differentiation is crucial for solving calculus problems involving exponential growth or decay, common in scientific fields.

Power Rule

The power rule is a fundamental principle in calculus that simplifies the process of differentiation, especially for polynomial functions. It states that for any function \( f(x) = x^n \), its derivative is given by \( \frac{d}{dx} x^n = nx^{n-1} \). Here, we find the derivative by:

• Multiplying the exponent \( n \) with the coefficient.
• Reducing the exponent by one.

Let's apply this to the term \( 4x^2 \) in our function. Using the power rule:

• Notice that the exponent is 2.
• Multiply the coefficient by the exponent: \( 2 \times 4x^{2-1} = 8x \).

This shows how the power rule quickly converts polynomial terms into their derivatives, making it easier to find slopes of curves and rates of change.

Derivative Calculation

Calculating the derivative of a function \( f(x) \) involves combining the derivatives of each term in the function. Once individual derivatives are found, they must be added together to form the final derivative expression.

• Begin by finding the derivative of each component term separately.
• Ensure to correctly apply differentiation rules to each term.
• Combine the derivatives of these terms to form \( f'(x) \).

For example, for \( f(x) = 2^{4x} + 4x^2 \), we first differentiated each term:

• The exponential term resulted in \( 4 \times 2^{4x} \ln(2) \).
• The polynomial term became \( 8x \).

Adding these results, we arrive at the combined derivative: \( f'(x) = 4 \times 2^{4x} \ln(2) + 8x \). Each step of the process requires attention to detail, but with practice, derivative calculations become increasingly intuitive.