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Chapter 2: Problem 3

Find the derivative of each function. $$f(x)=x^{500}$$

Short Answer

Expert verified
The derivative of \( f(x) = x^{500} \) is \( 500x^{499} \).

Step by step solution

01

Identify the Function

The function you are asked to differentiate is given as \( f(x) = x^{500} \). This is a power function where the base is \( x \) and the exponent is 500.
02

Recall the Power Rule for Derivatives

The power rule for derivatives states that for any function \( f(x) = x^n \), the derivative \( f'(x) \) is \( nx^{n-1} \). We will apply this rule to differentiate \( f(x) = x^{500} \).
03

Apply the Power Rule

Using the power rule, differentiate the function: \[ f'(x) = 500x^{500 - 1} = 500x^{499} \] This means you multiply the exponenet 500 by the original term and decrease the exponent by 1.
04

Simplify the Derivative

The derivative \( f'(x) = 500x^{499} \) is already in its simplest form since there are no coefficients to combine or other terms to simplify further.

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Key Concepts

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

Power Rule
The power rule is an essential tool for solving differentiation problems in calculus. It's specifically used to find the derivative of a power function, which is a function in the form of \( f(x) = x^n \). Using the power rule makes the process straightforward:
  • Identify the exponent \( n \) of the function \( x^n \).
  • Take this exponent and multiply it by \( x \).
  • Subtract one from the exponent to obtain the new power of \( x \).
These steps make it clear that for \( f(x) = x^{500} \), the derivative \( f'(x) \) is calculated as \( 500x^{499} \). The exponent 500 is now a coefficient, simplified through basic arithmetic.
Derivative of a Function
The derivative of a function represents its rate of change or how the function's output value changes as its input value changes. Differentiation, which is the process of finding the derivative, is fundamental in calculus and helps analyze real-world situations like velocity or rates of change in various contexts.

In the given exercise, the function \( f(x) = x^{500} \) was differentiated to understand how the function grows as \( x \) increases. The derivative, \( 500x^{499} \), tells us that for each increment in \( x \), the function increases by an amount proportional to \( 500x^{499} \). This insight is critical when dealing with polynomial functions, and the derivative offers a way to explore the behavior of these graphs.
Mathematical Notation
Mathematical notation is a language of symbols used to represent mathematical ideas clearly and concisely. In calculus, notation serves as a communication tool to express operations such as differentiation, easily understood within the mathematical community.

For derivatives, the most common notations include \( f'(x) \) and \( \frac{df}{dx} \). In our exercise example, \( f'(x) = 500x^{499} \) denotes the derivative of \( f(x) = x^{500} \). Here
  • \( f'(x) \) indicates the derivative of function \( f(x) \).
  • The power notation \( x^{499} \) signifies decreasing the exponent by one as per the power rule.
  • Simple arithmetic operations indicate how the differentiation process is performed.
Understanding these symbols can greatly enhance problem-solving efficiency and comprehension in calculus.

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Most popular questions from this chapter

A company's cost function is \(C(x)=\sqrt{4 x^{2}+900}\) dollars, where \(x\) is the number of units. Find the marginal cost function and evaluate it at \(x=20\).

Use the Generalized Power Rule to find the derivative of each function. $$y=\left(4-x^{2}\right)^{4}$$

Increasing global temperatures raise sea levels by thermal expansion and the melting of polar ice. Precise predictions are difficult, but a United Nations study predicts a rise in sea level (above the 2000 level of \(L(x)=0.02 x^{3}-0.07 x^{2}+8 x\) centimeters, where \(x\) is the number of decades since 2000 (so, for example, \(x=2\) means the year 2020). Find \(L(10), L^{\prime}(10)\), and \(L^{\prime \prime}(10)\), and interpret your answers. [Note: Rising sea levels could flood many islands and coastal regions.]

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Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$\left(\frac{x+1}{x-1}\right)^{4}$$
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