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Chapter 2: Problem 5

Find \(d y / d x\) $$ y=\pi^{3} $$

Short Answer

Expert verified
The derivative \( \frac{dy}{dx} = 0 \).

Step by step solution

01

Identify the function type

The function given is a constant function, where \( y = \pi^3 \). This means that no matter the value of \( x \), the value of \( y \) remains \( \pi^3 \).
02

Apply the derivative of a constant rule

The derivative of a constant is always zero. This is because a constant function does not change, and hence, the rate of change (or the slope) is zero. Mathematically, if \( y = c \) where \( c \) is a constant, then the derivative \( \frac{dy}{dx} = 0 \).
03

Calculate the derivative

Using the rule from Step 2, \( \frac{dy}{dx} \) is determined as follows: Since \( y = \pi^3 \) is a constant, \( \frac{dy}{dx} = 0 \).

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Key Concepts

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

Understanding Constant Function
In mathematics, a constant function is a function whose output value is the same for every input value. This means no matter what number you choose for the variable (often "x"), the output remains unchanged. For example, in the function given in the exercise,
  • \( y = \pi^3 \) is a constant function.
  • Here, \( \pi^3 \) is a numerical value that does not depend on the variable \( x \).
  • This implies the graph of a constant function is a horizontal line in the coordinate plane.
Therefore, no matter what x-coordinate you choose, the y-coordinate will always be \( \pi^3 \). Constant functions are straightforward, making them an excellent starting point for learning derivatives.
Rate of Change in Constant Functions
The concept of rate of change is central in calculus, particularly when calculating derivatives. It measures how a quantity changes in relation to another quantity. When we talk about the rate of change in the context of functions, we generally mean the slope of the function's graph. The slope is the measure of steepness or the degree of curvature in the graph.
  • For a constant function like \( y = \pi^3 \), the rate of change is zero.
  • Graphically, it translates to a flat line, indicating no increase or decrease as x changes.
Since there is no variation in input-output pairs, there's no 'movement' or 'change' to account for. That's why constant functions have a rate of change (or derivative) of zero.
Derivative Rules for Constant Functions
Derivative rules are guidelines that help us find derivatives without long calculations. One of the simplest rules is the derivative of a constant function.
  • For any constant \( c \), the derivative \( \frac{d}{dx}(c) = 0 \).
  • This is because the slope or rate of change of a constant function is zero.
  • No matter what value x takes, the derivative of a constant remains unchanged at zero.
The exercise perfectly applies this rule; once we identify the function \( y = \pi^3 \) as constant, we instantly know its derivative is zero. This rule is foundational and emphasizes that without change in the dependent variable, the derivative must be zero.

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Most popular questions from this chapter

Find the indicated derivative. $$ \lambda=\left(\frac{a u+b}{c u+d}\right)^{6} ; \text { find } \frac{d \lambda}{d u} \quad(a, b, c, d \text { constants }) $$

Find an equation for the tangent line to the graph at the specified value of \(x .\) $$ y=\sin \left(1+x^{3}\right), x=-3 $$

Show that the segment of the tangent line to the graph of \(y=1 / x\) that is cut off by the coordinate axes is bisected by the point of tangency.

Find \(f^{\prime}(x)\) $$ f(x)=\sqrt{4+\sqrt{3 x}} $$

Find the indicated derivative. $$ x=\csc ^{2}\left(\frac{\pi}{3}-y\right) ; \text { find } \frac{d x}{d y} $$
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