Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x}\left(10^{\left(x^{2}\right)}+\left(x^{2}\right)^{10}\right) $$

Step-by-Step Solution

Answer

Derivative is \( 2x \ln(10) \cdot 10^{x^2} + 20x(x^2)^9 \).

Step 1: Understand the Function

The function we need to differentiate is \( f(x) = 10^{x^2} + (x^2)^{10} \). This function consists of two terms: an exponential term \( 10^{x^2} \) and a power term \( (x^2)^{10} \).

Step 2: Differentiate the First Term

We need to find the derivative of \( 10^{x^2} \). Using the chain rule for exponential functions, the derivative is \( \ln(10) \cdot 10^{x^2} \cdot \frac{d}{dx}(x^2) \). The derivative of \( x^2 \) is \( 2x \). Thus, the derivative of the first term is \( \ln(10) \cdot 10^{x^2} \cdot 2x \).

Step 3: Differentiate the Second Term

Now, differentiate \( (x^2)^{10} \) using the power rule for functions. Let \( u = x^2 \), then \( u^{10} \) has the derivative \( 10u^9 \cdot \frac{d}{dx}(x^2) \). The derivative of \( x^2 \) is \( 2x \). Therefore, the derivative of the second term is \( 10(x^2)^9 \cdot 2x \).

Step 4: Combine the Derivatives

Add the derivatives from Steps 2 and 3 to obtain the overall derivative. The derivative is \( D_x(f(x)) = \ln(10) \cdot 10^{x^2} \cdot 2x + 10(x^2)^9 \cdot 2x \).

Step 5: Simplify the Expression

Simplify the expression from Step 4. The overall derivative is \( 2x \ln(10) \cdot 10^{x^2} + 20x(x^2)^9 \).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to find the derivative of composite functions. A composite function is one where another function is nested inside it. When you have a function within a function, you use the chain rule to differentiate it efficiently.

For example, consider our function's first term, \( 10^{x^2} \). It's a composite function because it can be seen as an exponential function \( 10^u \) where \( u = x^2 \). To differentiate such a function using the chain rule, follow these steps:

• Differentiate the outer function: Here, the outer function is \( 10^u \). The derivative with respect to \( u \) is \( \ln(10) \cdot 10^u \).
• Differentiate the inner function: The inner function is \( x^2 \), with its derivative being \( 2x \).
• Multiply them together: The result becomes \( \ln(10) \cdot 10^{x^2} \cdot 2x \).

By utilizing the chain rule, you can handle the complexity of nested functions and obtain their derivatives accurately.

Power Rule

The power rule is one of the simplest and most frequently used derivative rules, which states that for any function \( x^n \), the derivative is \( nx^{n-1} \). It's perfect for polynomial terms and helps simplify the differentiation process.

In our exercise, the second term \( (x^2)^{10} \) can be differentiated easily using the power rule. Here's how it works:

• Identify the exponent: The term \( (x^2)^{10} \) can be seen as a power function. Set \( u = x^2 \). Then, the function becomes \( u^{10} \).
• Apply the power rule: Differentiate \( u^{10} \), which gives \( 10u^9 \). Remember, this is with respect to \( u \).
• Find the derivative of \( u \): The derivative of \( x^2 \) with respect to \( x \) is \( 2x \).
• Combine using chain rule: Multiply these results together: \( 10(x^2)^9 \cdot 2x \).

Employing the power rule simplifies the process for such terms, quickly leading to the correct derivative.

Exponential Functions

Exponential functions are functions where the variable itself is in the exponent. In calculus, they frequently appear in various forms and require special attention when finding their derivatives, especially if the base of the exponent is a constant different from the natural number \( e \).

For our given function, the exponential term is \( 10^{x^2} \). When differentiating such functions, it’s vital to remember the following:

• Base constant not \( e \): When the base \( a \) is not \( e \), the derivative is calculated as \( a^{u} \times \ln(a) \times \frac{du}{dx} \), where \( u \) is the exponent function.
• Handle internally nested functions: In \( 10^{x^2} \), consider \( x^2 \) as \( u \). Hence, differentiate \( x^2 \) to get \( 2x \).
• Combine results: The derivative will be \( \ln(10) \cdot 10^{x^2} \cdot 2x \).

Understanding these aspects of exponential functions is crucial for differentiating them correctly, forming the backbone of calculus involving exponential terms.