Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated derivative. \(D_{x}\left(10^{\left(x^{2}\right)}+\left(x^{2}\right)^{10}\right)\)

Step-by-Step Solution

Answer

The derivative is \( D_x\left(10^{(x^2)} + (x^2)^{10}\right) = 10^{(x^2)} \cdot \ln(10) \cdot 2x + 20x(x^2)^9 \).

Step 1: Identify the function parts

The given function is \( f(x) = 10^{(x^2)} + (x^2)^{10} \). It consists of two parts: \( 10^{(x^2)} \) and \( (x^2)^{10} \) that we need to differentiate separately.

Step 2: Differentiate the first part

Apply the chain rule to differentiate \( 10^{(x^2)} \). The derivative of \( b^u \) with respect to \( x \) is \( b^u \ln(b) \frac{du}{dx} \). For \( u = x^2 \), the derivative is:\[ \frac{d}{dx}[10^{(x^2)}] = 10^{(x^2)} \cdot \ln(10) \cdot 2x \]

Step 3: Differentiate the second part

Apply the chain rule and power rule to differentiate \( (x^2)^{10} \). The derivative of \( u^n \) is \( n \cdot u^{n-1} \cdot \frac{du}{dx} \). For \( u = x^2 \), the derivative is:\[ \frac{d}{dx}[(x^2)^{10}] = 10 \cdot (x^2)^9 \cdot 2x = 20x(x^2)^9 \]

Step 4: Combine the derivatives

Add the derivatives from the previous steps to get the full derivative of the original function:\[ D_x\left(10^{(x^2)} + (x^2)^{10}\right) = 10^{(x^2)} \cdot \ln(10) \cdot 2x + 20x(x^2)^9 \]

Key Concepts

Chain Rule

In calculus differentiation, the chain rule is a key concept that helps us find the derivative of composite functions. Composite functions are functions that are nested inside each other, such as when one function’s output becomes another function’s input. The chain rule states that if you have a function that is composed of two functions, say \( f(g(x)) \), then the derivative of this composite function is the derivative of the outer function, \( f \), evaluated at \( g(x) \), multiplied by the derivative of the inner function, \( g(x) \). Mathematically, if \( y = f(g(x)) \), the chain rule tells us:\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]In this exercise, the chain rule is applied to both parts of the function. For the term \( 10^{(x^2)} \), the outer function is \( b^u \) and the inner function is \( x^2 \). Differentiating requires us to multiply the derivative of the outer function \( 10^{x^2} \cdot \ln(10)\) by the derivative of the inner function \( 2x \). Using the chain rule correctly helps in solving complex derivative problems where functions are layered within one another.

Power Rule

The power rule is another fundamental technique in calculus differentiation that simplifies finding the derivative of functions raised to a power. When a function \( x^n \) has a variable raised to a constant power, the power rule states that the derivative is found by multiplying the power by the function, and reducing the power by one:\[ \frac{d}{dx}[x^n] = n \cdot x^{n-1} \]In our given problem, for the part \( (x^2)^{10} \), the power rule can be applied effectively. Here, the base \( x^2 \) is raised to the power of 10. We simplify by bringing down the power 10, multiplying by the original \( x^{2}\), and reducing the power by one. Thus, it becomes \( 10 \cdot (x^2)^9 \), and because \( x^2 \) is not just \( x \), we additionally multiply by the derivative of \( x^2 \) using our friend, the chain rule, resulting in:\[ 10 \cdot (x^2)^9 \cdot 2x = 20x(x^2)^9 \]The power rule helps to quickly and effectively differentiate polynomial functions and others that involve variables raised to exponents.

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent, such as \( a^x \). Their differentiation involves exponential rules that are slightly different from polynomial rules.One of the main characteristics of exponential functions is that their rate of growth can be rapid, due to the variable being in the exponent.The derivative of an exponential function, such as \( b^u \), involves the natural logarithm of the base. This results in a formula:\[ \frac{d}{dx}[b^u] = b^u \cdot \ln(b) \cdot \frac{du}{dx} \]Here, \( b \) is the base, and \( u \) is a function of \( x \). In the exercise, \( 10^{x^2} \) exemplifies an exponential function where the base \( 10 \) is raised to the power of \( x^2 \). Differentiating it involves the derivative given by \( 10^{x^2} \cdot \ln(10) \cdot 2x \).Exponential functions are seen in a wide range of applications, from modeling population growth, radioactive decay, to financial growth over time. Understanding how to differentiate them is crucial for tackling complex calculus problems involving exponential change.