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Chapter 1: Problem 36

$$ \text { Differentiate. } $$ $$ \text { 36. } y=\pi^{2} x $$

Short Answer

Expert verified
The derivative of \(y = \pi^2 x\) is \(y' = \pi^2\).

Step by step solution

01

Identify the Rule for Differentiation

The given function is a linear function of the form \(y = kx\), where \(k\) is a constant. To differentiate this, apply the rule \(\frac{d}{dx}[kx] = k\).
02

Apply the Differentiation Rule

In this problem, \(k\) is \(\pi^2\). So, replace \(k\) with \(\pi^2\) in the differentiation rule: \[\frac{d}{dx} [\pi^2 x] = \pi^2\].
03

Write the Derivative

The derivative of the function \(y = \pi^2 x\) is therefore \(y' = \pi^2\).

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Key Concepts

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

Linear Function
A linear function is a type of function that creates a straight line when graphed. It has the form \(y = kx + c\) where \(k\) is the slope and \(c\) is the y-intercept. In the given exercise, the function \(y= \pi^{2} x\) is linear. Here, \(k = \pi^2\) and \(c = 0\), signaling that the line passes through the origin. Linear functions have a constant rate of change. This means that for each unit increase in \(x\), \(y\) increases by a fixed amount \(k\).
Differentiation Rule
Differentiation is a mathematical process used to find the rate of change of a function. The basic differentiation rule for a function of the form \(y = kx\) is given by \(\frac{d}{dx}[kx] = k\). This rule tells us that to differentiate a linear function, we simply take the constant multiplier \(k\) in front of \(x\). For example, given \(y = \pi^2 x\), applying this rule directly yields the derivative \(y' = \pi^2\). This means the rate of change of the function with respect to \(x\) is \pi^2.
Constant Multiplier Rule
The constant multiplier rule is a useful principle in differentiation. It states that when you have a constant multiplied by a function, you can

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Most popular questions from this chapter

Compute the following. . \(f^{\prime}(1)\) and \(f^{\prime \prime}(1)\), when \(f(t)=\frac{1}{2+t}\)

Use limits to compute \(f^{\prime}(x)\). [Hint: In Exercises \(45-48\), use the rationalization trick of Example \(8 .]\) \(f(x)=\frac{x}{x+2}\)

Each limit in Exercises 49-54 is a definition of \(f^{\prime}(a)\). Determine the function \(f(x)\) and the value of \(a\). \(\lim _{h \rightarrow 0} \frac{(64+h)^{1 / 3}-4}{h}\)

Let \(R(x)\) denote the revenue (in thousands of dollars) generated from the production of \(x\) units of computer chips per day, where each unit consists of 100 chips. (a) Represent the following statement by equations involving \(R\) or \(R^{\prime}\) : When 1200 chips are produced per day, the revenue is \(\$ 22,000\) and the marginal revenue is \(\$ .75\) per chip. (b) If the marginal cost of producing 1200 chips is \(\$ 1.5\) per chip, what is the marginal profit at this production level?

In Exercises 87-90, use paper and pencil to find the equation of the tangent line to the graph of the function at the designated point. Then, graph both the function and the line to confirm it is indeed the sought-after tangent line. \(f(x)=\sqrt{x},(9,3)\)
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