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Chapter 2: Problem 34

Find \(f^{\prime}(x)\) by using the definition of the derivative. \(f(x)=\pi\)

Short Answer

Expert verified
The derivative of \( f(x) = \pi \) is \( 0 \).

Step by step solution

01

Understand the Function

The function given is a constant function, where \( f(x) = \pi \). This means for every value of \( x \), \( f(x) \) is always \( \pi \).
02

Recall the Definition of the Derivative

The derivative of a function \( f(x) \) at a point \( x \) is defined as: \[ f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
03

Substitute the Function into the Derivative Formula

Substitute the constant function \( f(x) = \pi \) into the derivative formula: \[ f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{\pi - \pi}{h} \]
04

Simplify the Expression

Simplify the expression inside the limit: \[ f^{\prime}(x) = \lim_{h \to 0} \frac{0}{h} \] Since \( \pi - \pi = 0 \).
05

Calculate the Limit

Evaluate the limit: \[ f^{\prime}(x) = \lim_{h \to 0} 0 = 0 \]
06

Conclusion

The derivative of a constant function is always 0. Therefore, for \( f(x) = \pi \), \( f^{\prime}(x) = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant Function
A constant function is one of the simplest types of functions in calculus. It is a function where a single fixed value, called the constant, is assigned to every input value:
  • If you take any value for the variable, the function's output remains unchanged.
  • In mathematical terms, for a constant function expressed as \( f(x) = c \), the output \( c \) stays the same for all \( x \).
In the exercise, we have \( f(x) = \pi \), which is a constant as \( \pi \) is a known numerical value.
In every case, regardless of what \( x \) represents, \( f(x) \) will always equal \( \pi \). This permanence is why the derivative of constant functions is key in calculus analysis.
Definition of the Derivative
The derivative of a function provides insight into how a function's output changes in relation to changes in its input.
  • It can be thought of as the function's rate of change or slope at any given point.
  • The formal definition involves a limit as it approaches a crucial mathematical boundary where \( h \) represents an infinitely small step toward zero.
The formula, \[f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]represents the derivative. It basically measures how much \( f(x) \) changes as \( x \) changes by a tiny amount \( h \). For our constant function \( f(x) = \pi \), substituting into this formula shows why its derivative equals zero: because there is no change as \( h \) approaches zero.
Limits in Calculus
Limits are foundational in calculus, acting as a bridge to understanding continuity, derivatives, and integrals. Limits are used to describe what happens to a function's output as the input gets arbitrarily close to a particular point.
  • They capture behavior around points that might be problematic directly.
  • For derivatives, limits help define the precise rate of change at a single point.
In the example of the derivative calculation, \[f^{\prime}(x) = \lim_{h \to 0} \frac{\pi - \pi}{h} = \lim_{h \to 0} \frac{0}{h} = 0\] we observe how \( \frac{0}{h} \) remains zero as \( h \) goes to zero, confirming that the behavior is consistent at, before, and after the point. Hence, limits play a crucial role in rigorously determining how functions smooth out, spike, or flatten, aiding in the understanding of their derivatives.

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Most popular questions from this chapter

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