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Chapter 1: Problem 36

Find \(f^{\prime}(x)\). $$f(x)=\frac{3 x}{4}$$

Short Answer

Expert verified
The derivative $${f}^{\text{'}}(x)$$ is $$\frac{3}{4}.$$

Step by step solution

01

Identify the Function

The given function is $$f(x)=\frac{3 x}{4}.$$
02

Apply the Constant Multiple Rule

The Constant Multiple Rule states that if a function is multiplied by a constant, the derivative of the function is the constant multiplied by the derivative of the function. Here, the constant is $$\frac{3}{4}.$$
03

Differentiate the Simplified Function

First, treat $$f(x)=\frac{3}{4}x$$ as $$f(x) = kx$$ where $$k$$ is a constant. Next, use the power rule to differentiate $$kx.$$ The derivative of $$x$$ is $$1.$$ Consequently, the derivative of $$kx$$ is $$k \times 1.$$
04

Combine Results

So, the derivative $${f}^{\text{'}}(x)$$ is $$\frac{3}{4} \times 1 = \frac{3}{4}.$$

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Key Concepts

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

Constant Multiple Rule
The constant multiple rule is a handy principle when dealing with derivatives. If a function is multiplied by a constant, the derivative of the function will be the constant times the derivative of the function without the constant.

For example, consider the function \( f(x) = c \times g(x) \), where \( c \) is a constant. According to the constant multiple rule:

  • The derivative of \( f(x) \) can be written as \( f'(x) = c \times g'(x) \).

To understand this better, let's look at our specific function: \( f(x) = \frac{3}{4} x \). Here, the constant \( c \) is \( \frac{3}{4} \). Using the constant multiple rule, the derivative is: \( f'(x) = \frac{3}{4} \times ( \text{derivative of } x ) \).
Power Rule
The power rule is one of the most fundamental rules for finding derivatives. It states that if you have a function in the form \( x^n \), where \( n \) is any real number, its derivative is \( n \times x^{n-1} \).

This can be expressed as:
  • \( \frac{d}{dx} x^n = n \times x^{n-1} \)

In our example function \( f(x) = \frac{3}{4} x \), the term \( x \) can be seen as \( x^1 \). Applying the power rule here:
  • \( \frac{d}{dx} x^1 = 1 \times x^{1-1} = 1 \times x^0 = 1 \).

It's important to note that any number to the power of zero is 1.
Differentiation
Differentiation is the process of finding the derivative of a function. The derivative is a measure of how a function changes as its input changes. In simpler terms, it gives us the rate of change or the slope of the function at any given point.

To differentiate a function, follow these steps:
  • Identify the function you need to differentiate.
  • Apply the relevant differentiation rules, such as the power rule, constant multiple rule, product rule, or chain rule.
  • Simplify the derivative obtained.

For our specific example function \( f(x) = \frac{3}{4}x \):
  • We identified the function.
  • Applied the constant multiple rule to isolate the constant.
  • Used the power rule to find the derivative of \( x \).
  • Then multiplied the constant \( \frac{3}{4} \) with the derivative of \( x \).
Combining all these steps, feel confident that the derivative of \( f(x) \) is indeed \( \frac{3}{4} \).

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Most popular questions from this chapter

Is the derivative of the reciprocal of \(f(x)\) the reciprocal of the derivative of \(f^{\prime}(x) ?\) Why or why not?

Find \(d y / d x .\) Each function can be differentiated using the rules developed in this section, but some algebra may be required beforehand. $$y=(\sqrt{x}+\sqrt[3]{x})^{2}$$

Utility is a type of function that occurs in economics. When a consumer receives \(x\) units of a product, a certain amount of pleasure, or utility, \(U\), is derived. Suppose that the utility related to the number of tickets \(x\) for a ride at a county fair is $$U(x)=80 \sqrt{\frac{2 x+1}{3 x+4}}$$ Find the rate at which the utility changes with respect to the number of tickets bought.

Compound interest. If \$ 1000\( is invested at interest rate i, compounded annually, in 3 yr it will grow to an amount A given by (see Section R.1) \)A=\$ 1000(1+i)^{3}. a) Find the rate of change, \(d A / d i\) B) Interpret the meaning of dA/di.

First, use the Chain Rule to find the answer. Next, check your answer by finding \(f(g(x))\) taking the derivative, and substituting. \(f(u)=\sqrt[3]{u}, g(x)=u=1+3 x^{2}\) Find \((f \circ g)^{\prime}(2)\)
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