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Chapter 3: Problem 1

Find the derivatives of the given functions. \(x^{100} \)

Short Answer

Expert verified
The derivative of \(x^{100}\) is \(100x^{99}\).

Step by step solution

01

Identify the Function and the Rule

The given function is \(f(x) = x^{100}\). To find its derivative, we will use the Power Rule for derivatives. The Power Rule states that if \(f(x) = x^n\), then the derivative, \(f'(x)\), is \(n \cdot x^{n-1}\).
02

Apply the Power Rule

Apply the Power Rule to find the derivative of \(f(x) = x^{100}\). According to the rule, the derivative is \(f'(x) = 100 \cdot x^{100-1}\).
03

Simplify the Expression

Simplify the expression obtained from applying the Power Rule. The expression \(100 \cdot x^{100-1}\) simplifies to \(100x^{99}\). Therefore, the derivative of the function \(x^{100}\) is \(100x^{99}\).

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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 finding derivatives, especially in single variable calculus. It allows us to differentiate functions that are powers of a variable. The rule is straightforward: for a function of the form \( f(x) = x^n \), its derivative is \( f'(x) = n \cdot x^{n-1} \). This means we multiply the exponent by the variable raised to the one less than its original power.

Let's break this down:
  • Start with a function that involves a power of \( x \). For instance, \( x^{100} \).
  • Identify the exponent, which is \( 100 \) in this case.
  • Apply the Power Rule: multiply the exponent by the variable \( x \) raised to the power of the exponent minus one.
The beauty of the Power Rule is how it simplifies tasks that would otherwise involve complex algebraic manipulations. This is why it's so widely used in calculus.
Differentiation
Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function changes at any point. It's essentially the process of finding derivatives. Through differentiation, we can calculate the slope of a tangent line to the curve of a function at a given point.

When differentiating something like \( x^{100} \), we follow these straightforward steps:
  • Identify the type of function you have, such as a power function.
  • Use the appropriate differentiation rule, like the Power Rule, to find the derivative.
  • Simplify the expression to its most basic form, which in the case of our example is \( 100x^{99} \).
By mastering differentiation, we can understand how quantities change and predict behaviors in mathematical models.
Single Variable Calculus
Single Variable Calculus is the study of functions involving one variable and includes topics such as limits, derivatives, and integrals. It forms the foundation for more advanced calculus studies. In the realm of differentiation, single variable calculus focuses on finding the derivative of functions involving just one independent variable.

For example, with a function like \( f(x) = x^{100} \):
  • Identify \( x \), our single variable, as the base of analysis.
  • Use differentiation techniques, such as the Power Rule, which are particularly tailored for functions of this type.
  • Interpret the resulting derivative, \( 100x^{99} \), in terms of rates of change.
Mastering single variable calculus allows students to solve real-world problems and set the groundwork for multi-variable calculus and other advanced mathematical concepts.

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Most popular questions from this chapter

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible. \((x+8)^{5}\)

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible. \((4-x)^{3}\)

Use the product rule to compute the derivative of \(f(x)=(2 x-3)^{2}\). Sketch the function. Find an equation of the tangent line to the curve at \(x=2 .\) Sketch the tangent line at \(x=2\).

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible. \(x^{3}-2 x^{2}+4 \sqrt{x}\)

Find an equation for the tangent line to \(f(x)=(x-2) /\left(x^{3}+4 x-1\right)\) at \(x=1 \)
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