Problem 28 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The derivative is \(f'(t) = A e^t + \frac{B}{t}\).

Step 1: Differentiate the First Term

The first term of the function is \(A e^t\). To find its derivative, apply the basic rule that the derivative of \(e^t\) is still \(e^t\). Since \(A\) is a constant, the derivative of \(A e^t\) is \(A e^t\).

Step 2: Differentiate the Second Term

The second term is \(B \ln t\). To find its derivative, apply the rule that the derivative of \(\ln t\) is \(1/t\). Since \(B\) is a constant, the derivative of \(B \ln t\) is \(\frac{B}{t}\).

Step 3: Combine the Derivatives

Combine the derivatives of each term to determine the derivative of the entire function \(f(t) = A e^t + B \ln t\). The derivative \(f'(t)\) is: \[f'(t) = A e^t + \frac{B}{t}.\]

Key Concepts

Exponential Functions

Exponential functions are an important class of functions in calculus and many other areas of mathematics. An exponential function is typically written in the form \( f(x) = a e^{bx} \), where \(e\) is the base of the natural logarithm, approximately equal to 2.718. The variable \(x\) is the exponent in this case, and \(a\) and \(b\) are constants that scale and transform the function.
In calculus, exponential functions are unique because their rate of growth or decay is proportional to their current value. This means that when you differentiate an exponential function, the form of the derivative is very similar to that of the original function.

• The derivative of \(e^{x}\) is \(e^{x}\) itself. This property is especially useful because it simplifies the process of differentiation.
• When you have a constant multiplied by \(e^{x}\), like \(A e^{x}\), you apply the constant rule to keep \(A\) as it is, resulting in a derivative of \(A e^{x}\).

This simplicity and predictability make exponential functions a powerful tool in both theoretical and applied mathematics.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are integral in understanding growth patterns and calculating derivatives. Commonly, a logarithmic function is expressed as \( f(x) = \log_{b}(x) \) or \( f(x) = \ln(x) \), where \(\ln(x)\) represents the natural logarithm with base \(e\).
The natural logarithm, \(\ln(x)\), has the derivative \(1/x\). This is crucial when solving problems involving growth decay and rates.

• The derivative of \(\ln(t)\) is \(1/t\). This means that small changes in \(t\) result in small proportional changes in \(\ln(t)\).
• When dealing with a constant multiplied by a natural logarithm, such as \(B \ln(t)\), you would keep the constant outside and differentiate the logarithm separately, giving \(B/t\) as its derivative.

Understanding the derivative of logarithmic functions allows one to comprehend the nature of changes with respect to time or another variable, which is crucial in fields like physics, economics, and beyond.

Chain Rule

The Chain Rule is a fundamental concept in calculus used to differentiate composite functions. It's crucial for finding the derivative when a function is nested within another function. The general formula of the Chain Rule is expressed as:
\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]
Let's break this down:

• The outer function \(f\) is differentiated first, with its argument left intact.
• Then, the inner function \(g(x)\) is differentiated separately.
• Finally, the two derivatives are multiplied together.

While the Chain Rule doesn't directly apply to the exercise's function \(f(t) = A e^t + B \ln t\), since it's already a sum of functions rather than a composite one, learning it gives insight into its logical structure. When functions require more complex differentiation processes, the Chain Rule becomes indispensable. It is especially handy when dealing with reactions happening in sequence, such as calculating rates of change in physics or biology, thereby ensuring that we capture all elements affecting the rate of change.