Question

Compute the derivative of the given function. $$f(t)=\ln (17)+e^{2}+\sin \pi / 2$$

Short answer

The derivative is 0.

Step-by-step solution

Understanding the Components of the Function

First, we need to identify the components of the function that require differentiation. The function is given as: \( f(t) = \ln(17) + e^2 + \sin \frac{\pi}{2} \). These are constants in terms of \( t \), as there is no \( t \)-variable present in any part of the expression.

Differentiation of Constant Terms

When differentiating a constant function with respect to a variable, the derivative is always zero. Since \( \ln(17) \), \( e^2 \), and \( \sin \frac{\pi}{2} \) are all constants, the derivative of the entire function \( f(t) \) with respect to \( t \) will be zero.

Applying the Constant Rule for Derivatives

Using the constant rule for differentiation, which states that if \( c \) is a constant, then \( \frac{d}{dt}[c] = 0 \), we apply this to each term in \( f(t) \). As none of the terms contain \( t \), the derivative \( f'(t) \) results in \( 0 \).

Key concepts

Constant Function Differentiation

When tackling the derivative of a constant function, the process is incredibly straightforward. A constant function is any function that does not change as the variable changes. For example, functions like \( f(x) = 5 \) or \( f(x) = \ln(17) + e^2 + \sin \frac{\pi}{2} \) maintain the same value regardless of the input for \( x \) or \( t \).

A hallmark of constant function differentiation is that the derivative of any constant value is zero. This is because differentiation measures how a function changes with respect to changes in the variable, and since a constant function does not change, its rate of change is zero.

When taking the derivative of constant values, you are essentially asking: "How much does \( f \) change as \( x \) changes?" The answer is not at all, so the derivative is \( 0 \). Using this basic rule makes differentiating any constants a breeze.

Application of the Derivative

The application of the derivative is a crucial tool in the field of calculus and beyond. It allows us to understand and calculate the rate at which one quantity changes compared to another. In practical terms:

• Derivatives help in finding the slope of a curve at any given point, essentially giving the instantaneous rate of change.
• They are crucial for solving problems in physics, engineering, economics, and even biology, where understanding how quantities change over time or in response to other variables is vital.

In our specific exercise, the application of the derivative highlights how constant components in a function don't contribute to changes in the dependent variable.

Since the function \( f(t) \) lacks a variable 't' in any term (\( \ln(17) \), \( e^2 \), \( \sin \frac{\pi}{2} \)), applying the derivative helps confirm that they do not affect the outcome, reasserting that they are constants. Therefore, we see that derivatives serve as both a mathematical operation and as a broader concept, informing us about the behavior of different types of functions.

Basic Differentiation Rules

Understanding the basic differentiation rules is like learning the ABCs of calculus. These rules lay down the foundation for more complex concepts. Key rules include:

• Constant Rule: The derivative of any constant is zero.
• Power Rule: For any expression \( x^n \), the derivative is \( nx^{n-1} \).
• Sum Rule: The derivative of a sum is the sum of the derivatives.

These principles simplify the differentiation process and make it methodical.

By applying these rules consistently, we can tackle any differentiation problem.

In our solution, we saw the constant rule come into play directly. Each part of the given function \( f(t) = \ln(17) + e^2 + \sin\ (\pi/2) \), when differentiated, resulted in zero.

Mastery of these basic rules will empower you to confidently and accurately work through an array of functions and their derivatives in calculus.