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Chapter 1: Problem 27

Differentiate. $$y=3 x+\pi^{3}$$

Short Answer

Expert verified
The derivative is 3.

Step by step solution

01

Identify the function components

The given function is a linear combination of two terms: one involving the variable x and a constant term. The function is given as y = 3x + \pi^{3}where \(3x\) is the term involving the variable and \pi^{3} is a constant.
02

Differentiate the variable term

Differentiate the term that involves the variable x. The derivative of \(3x\) with respect to x is the constant coefficient.\[\frac{d}{dx}(3x) = 3\]
03

Differentiate the constant term

The derivative of a constant term is zero. Therefore, the derivative of \pi^{3} is zero.\[\frac{d}{dx}(\pi^{3}) = 0\]
04

Combine the derivatives

Combine the results from Steps 2 and 3 to get the final derivative. The combined result is:\[\frac{dy}{dx} = 3 + 0 = 3\]

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Key Concepts

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

derivative of a constant
When we talk about the derivative of a constant, we mean the rate at which the constant changes. Constants are fixed values, meaning they do not change regardless of the variable. Therefore, the derivative of a constant is always zero.

For example, in the function given in the exercise, \(\text{y = 3x + \pi^{3}}\), \(\text{\pi^{3}}\) is a constant term. To find its rate of change with respect to x, we differentiate it: \[\frac{d}{dx}(\pi^{3}) = 0\]
The reasoning is simple: since \(\text{\pi^{3}}\) does not vary as x changes, its derivative is zero.
linear function differentiation
Differentiation of linear functions involves finding the rate of change or the slope of the function. Linear functions are of the form \(\text{y = ax} + \text{b}\), where a and b are constants.

In our given function, \(\text{y = 3x + \pi^{3}}\), the term \(\text{3x}\) is linear. To differentiate a linear function, we use the power rule. The power rule states:
  • For \(\text{f(x) = x^{n}}\), \[\frac{d}{dx}(x^{n}) = nx^{n-1}\]
  • When n=1 (as in any linear term like \(\text{3x}\)), \[\frac{d}{dx}(x) = 1\]
    Then, multiply by the constant coefficient if present. Hence,
    • For \(\text{3x}\), \[\frac{d}{dx}(3x) = 3\]
      The coefficient 3 indicates how steep the line is.
fundamental theorem of calculus
The Fundamental Theorem of Calculus connects differentiation and integration, showing that they are inverse processes. There are two parts to this theorem:
  • First Part: It states that if \(\text{F(x)}\) is an antiderivative of \(\text{f(x)}\), then the integral of \(\text{f(x)}\) from \(\text{a to b}\) is given by \(\text{F(b) - F(a)}\).
  • Second Part: It tells us that the derivative of the integral function \(\text{F(x)}\) with respect to x gives back the original function \(\text{f(x)}\). It can be written as:
    \[\frac{d}{dx} \left( \int_{a}^{x} f(t) dt \right) = f(x)\]
    This theorem underlines the connection between finding areas (integration) and finding slopes (differentiation). It fundamentally shows how differential and integral calculus are bound together.

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Most popular questions from this chapter

Using the sum rule and the constant-multiple rule, show that for any functions \(f(x)\) and \(g(x).\) $$\frac{d}{d x}[f(x)-g(x)]=\frac{d}{d x} f(x)-\frac{d}{d x} g(x).$$

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=\sqrt{x^{2}+1}$$

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=x^{2}$$

Let \(f(p)\) be the number of cars sold when the price is \(p\) dollars per car. Interpret the statements \(f(10,000)=200,000\) and \(f^{\prime}(10,000)=-3\).

If \(h(x)=[f(x)]^{2}+\sqrt{g(x)},\) determine \(h(1)\) and \(h^{\prime}(1),\) given that \(f(1)=1, g(1)=4, f^{\prime}(1)=-1,\) and \(g^{\prime}(1)=4\).
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