Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated derivative. \(D_{x}\left[x^{\pi+1}+(\pi+1)^{x}\right]\)

Step-by-Step Solution

Answer

Derivative is \((\pi+1)x^\pi + (\pi+1)^x \ln(\pi+1)\)."

Step 1: Differentiate the Power Function

Begin by differentiating the function term by term. The first term is \(x^{\pi+1}\). Use the power rule \(\frac{d}{dx}[x^n] = nx^{n-1}\). In this case, \(n = \pi+1\). Differentiating gives: \(\pi+1) \cdot x^{\pi+1-1} = (\pi+1) \cdot x^{\pi}\).

Step 2: Differentiate the Exponential Function

Next, differentiate the second term \((\pi+1)^x\). Use the formula \(\frac{d}{dx}[a^x] = a^x \ln(a)\), where \(a = \pi+1\). Thus, the derivative is \((\pi+1)^x \ln(\pi+1)\).

Step 3: Combine the Derivatives

Sum the derivatives of both terms to give the overall derivative of the function. Thus, the derivative \(D_x[x^{\pi+1} + (\pi+1)^x]\) is \((\pi+1)x^\pi + (\pi+1)^x\ln(\pi+1)\).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus used to find the derivative of a function of the form \( x^n \), where \( n \) can be any real number. This rule is particularly powerful because it simplifies the process of differentiation to a straightforward formula. To apply the power rule, you simply bring the exponent down in front of the \( x \), and then reduce the exponent by one.For example, for \( x^n \), the derivative is calculated as follows:

• Multiply the exponent \( n \) by the variable \( x \) raised to the power of \( n-1 \).
• The resulting derivative is \( nx^{n-1} \).

In our exercise, we have the term \( x^{\pi + 1} \). Using the power rule, we take the exponent \( \pi + 1 \) and multiply it by \( x \) raised to the power of \( \pi \) (since \( \pi + 1 - 1 = \pi \)).This means the derivative for \( x^{\pi+1} \) is \((\pi+1) \cdot x^{\pi}\). Understanding the power rule can greatly ease the differentiation process in more complex calculus problems.

Exponential Function Differentiation

When differentiating exponential functions, the rule differs slightly from the power rule due to the nature of exponential growth. In this context, an exponential function can be represented as \(a^x\), where \(a\) is a constant and \(x\) is the variable in play.To find the derivative of an exponential function, you use the following formula:

• \( \frac{d}{dx}[a^x] = a^x \ln(a) \).
• This formula shows that the derivative of \( a^x \) includes the original function \( a^x \) multiplied by the natural logarithm of \( a \).

Applying this to our specific equation, we differentiate the term \( (\pi+1)^x \). Here, \( a \) is \( \pi+1 \) and the differentiation gives us \( (\pi+1)^x \ln(\pi+1) \).This rule is crucial when dealing with exponential functions, as it allows you to understand growth rates and changes in contexts like compound interest or population modeling.

Calculus

Calculus is the branch of mathematics that studies how things change. It's divided into two main areas: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which represents how a function changes as its input changes.In our exercise, calculus helps to determine the rate of change of a function expressed as \( D_{x}[x^{\pi+1} + (\pi+1)^{x}] \), by finding its derivative. This involves breaking down the function into separate components (like power and exponential functions) and applying rules to each.Some key points about calculus and why it's essential to grasp:

• Calculus allows us to find the velocity of an object at any given moment.
• It enables the computation of areas and volumes of irregular shapes.
• It is used in various fields, from physics to economics, to predict future trends and behaviors.

Understanding calculus provides critical problem-solving skills needed for advanced subjects, giving insights into the underlying principles governing not just mathematics, but many scientific and real-world phenomena.