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Chapter 4: Problem 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. $$x^{3}-2 x^{2}+4 \sqrt{x}$$

Short Answer

Expert verified
The derivative is \( f'(x) = 3x^2 - 4x + \frac{2}{\sqrt{x}} \).

Step by step solution

01

Identify the function

The function whose derivative we need to find is \( f(x) = x^3 - 2x^2 + 4\sqrt{x} \). This function is a combination of power functions and a square root function.
02

Apply the Power Rule

For the terms \( x^3 \) and \( -2x^2 \), we use the power rule for differentiation. The power rule states that for \( x^n \), the derivative is \( nx^{n-1} \). Thus, the derivative of \( x^3 \) is \( 3x^2 \), and the derivative of \( -2x^2 \) is \(-4x^1 = -4x\).
03

Differentiate the square root function

The term \( 4\sqrt{x} \) can be rewritten as \( 4x^{1/2} \). Using the power rule on this term, the derivative is \( (4 \cdot \frac{1}{2})x^{-1/2} = 2x^{-1/2} = \frac{2}{\sqrt{x}} \).
04

Combine derivatives

Add the derivatives of each term to get the derivative of the entire function: \( f'(x) = 3x^2 - 4x + \frac{2}{\sqrt{x}} \).

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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 a fundamental technique in calculus used to find the derivative of functions that are powers of a variable. This rule simplifies the process of differentiation by providing a straightforward formula: if you have a function in the form of \( x^n \), its derivative is \( nx^{n-1} \). This means you multiply the power by the coefficient and then decrease the power by one.
Examples make this concept clearer:
  • For \( f(x) = x^3 \), applying the power rule gives \( f'(x) = 3x^2 \).
  • If \( g(x) = -2x^2 \), then \( g'(x) = -4x \) after using the power rule.
This rule is not only a powerful tool for simple polynomials but can also be applied to more complex problems when differentiation is needed. Understanding the power rule is crucial for tackling a wide range of calculus problems.
Square Root Function
A square root function is a type of algebraic function involving the square root of a variable. In terms of differentiation, this can initially seem tricky. However, by reframing the function as a power of \( x \), it becomes easier to differentiate.
The term \( \sqrt{x} \) can be expressed as \( x^{1/2} \). This transformation allows us to apply the power rule to find its derivative. Applying the power rule:
  • Start with \( 4\sqrt{x} = 4x^{1/2} \).
  • The derivative becomes \( 4 \cdot \frac{1}{2}x^{-1/2} = 2x^{-1/2} \).
  • This can further be written as \( \frac{2}{\sqrt{x}} \).
Rewriting the square root in this way makes it much simpler to apply differentiation techniques, allowing us to handle these functions with ease.
Differentiation Techniques
Differentiation is a cornerstone of calculus, involving the process of finding a function's rate of change. Various techniques exist to differentiate different types of functions. Some of the most useful techniques include product rule, chain rule, and our focus here: the power rule.
To differentiate the given function \( f(x) = x^3 - 2x^2 + 4\sqrt{x} \), we apply different techniques for each term involved:
  • For power functions like \( x^3 \) and \( -2x^2 \), we use the simple power rule.
  • The square root function \( 4\sqrt{x} \) is rewritten and differentiated using the power rule.
  • After differentiating each term, you combine all derivatives to get the overall derivative: \( f'(x) = 3x^2 - 4x + \frac{2}{\sqrt{x}} \).
Understanding when and how to use these techniques will enhance your ability to tackle a wide range of problems efficiently. Each problem might require using different rules, or even a combination, but the basics remain the same.

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Most popular questions from this chapter

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