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Chapter 1: Problem 6

$$ \text { Differentiate. } $$ $$ f(x)=12+\frac{1}{7^{3}} $$

Short Answer

Expert verified
The derivative of the function is 0

Step by step solution

01

Identify Constant Terms

The function is given by \( f(x) = 12 + \frac{1}{7^3} \). Notice that both 12 and \( \frac{1}{7^3} \) are constants.
02

Recall the Derivative of a Constant

The derivative of any constant is zero. Write this down for clarity: \( \frac{d}{dx}(C) = 0 \) where C is a constant.
03

Apply the Derivative Rule

Differentiate each term of the function separately: \[ f'(x) = \frac{d}{dx}(12) + \frac{d}{dx}\left( \frac{1}{7^3} \right) \]
04

Compute the Derivatives

Since both 12 and \( \frac{1}{7^3} \) are constants, their derivatives are zero: \[ f'(x) = 0 + 0 = 0 \]

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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 a type of function where the output value is the same regardless of the input value. It can be written as \( f(x) = C \), where \( C \) is a constant. For example, in the exercise given, we have a constant function where \( f(x) = 12 + \frac{1}{7^3} \). Here, both 12 and \( \frac{1}{7^3} \) are constants.

Key characteristics of constant functions:
  • The graph of a constant function is a horizontal line.
  • No matter what value of \( x \) you input, the output remains the same.
  • They do not change, hence their name 'constant.'
Differentiation
Differentiation is a fundamental tool in calculus. It is the process of finding the derivative of a function. The derivative represents how a function changes as its input changes. In simpler terms, it helps us understand the rate at which something is changing.

For constants:
  • The derivative of a constant is always zero, because constants do not change.
  • For instance, if \( C \) is a constant, then \( \frac{d}{dx}(C) = 0 \).
In our exercise, since both 12 and \( \frac{1}{7^3} \) are constants, their derivatives are zero. This is why when we differentiate, the result is:
\[ f'(x) = \frac{d}{dx}(12) + \frac{d}{dx}\left( \frac{1}{7^3} \right) = 0 + 0 = 0 \]

Remember, differentiation helps to find how a function changes, but if the function does not change, its derivative is zero.
Basic Calculus Rules
In basic calculus, there are several fundamental rules we use to simplify differentiation. Some of these include:
  • Constant Rule: The derivative of any constant is zero.
  • Sum Rule: The derivative of the sum of two functions is the sum of their derivatives.
    Example: If \( f(x) = g(x) + h(x) \), then \( f'(x) = g'(x) + h'(x) \).

In the given exercise, both rules are applied when differentiating the function \( f(x) = 12 + \frac{1}{7^3} \). Since both terms are constants, we use the constant rule and get zero. By the sum rule, we add up the derivatives to obtain:

\[ f'(x) = 0 + 0 = 0 \]

These rules make complex functions easier to work with by breaking them down into simpler parts that we can handle easily.

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Most popular questions from this chapter

Suppose that \(f(t)=3 t+2-\frac{12}{t}\) (a) What is the average rate of change of \(f(t)\) over the interval 2 to 3 ? (b) What is the (instantaneous) rate of change of \(f(t)\) when \(t=2 ?\)

Determine the value of \(a\) that makes the function \(f(x)\) continuous at \(x=0\). \(f(x)=\left\\{\begin{array}{ll}2(x-a) & \text { for } x \geq 0 \\ x^{2}+1 & \text { for } x<0\end{array}\right.\)

Let \(f(x)\) be the number (in thousands) of computers sold when the price is \(x\) hundred dollars per computer. Interpret the statements \(f(12)=60\) and \(f^{\prime}(12)=-2 .\) Then, estimate the number of computers sold if the price is set at \(\$ 1250\) per computer.

Compute the following. \(g^{\prime}(0)\) and \(g^{\prime \prime}(0)\), when \(g(T)=(T+2)^{3}\)

Find the indicated derivative. \(\frac{d y}{d x}\) if \(y=1\)
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