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Chapter 3: Problem 20

Find the first derivative. $$ p(x)=\frac{2 x^{4}+3 x^{2}-1}{x^{2}} $$

Short Answer

Expert verified
The first derivative is \(p'(x) = 4x - 2x^{-3}.\)

Step by step solution

01

Simplifying the Expression

Before finding the derivative, simplify the function by dividing each term in the numerator by the denominator \(x^2\). This gives:\[ p(x) = 2x^2 + 3 - x^{-2}. \]
02

Applying the Power Rule

Use the power rule for derivatives, which states \( \frac{d}{dx}[x^n] = nx^{n-1} \). Apply it to each term:- For \(2x^2\), the derivative is \(4x\).- For \(3\), the derivative is \(0\), since the derivative of a constant is zero.- For \(-x^{-2}\), the derivative is \(2x^{-3}\) (negative sign becomes positive because of the negative exponent).So, the first derivative is:\[ p'(x) = 4x + 0 - 2x^{-3}. \]
03

Writing the Final Derivative

Combine the terms obtained in the previous step to write the final expression for the derivative:\[ p'(x) = 4x - 2x^{-3}. \] This is the first derivative of the function \(p(x)\).

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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 concept in calculus that simplifies the differentiation of polynomial functions. It's essential for finding derivatives efficiently.To apply the power rule, remember:
  • The derivative of a function of the form \( x^n \) is \( nx^{n-1} \).
Consider this mechanism for each term separately:
  • If you have \( 2x^2 \), its derivative is \( 4x \). Here, you multiply 2 (the coefficient) by the exponent 2, and subtract one from the exponent.
  • For a constant like 3, the derivative is 0, as constants don鈥檛 change.
  • If you have a term like \(-x^{-2}\), the negative exponent is handled the same way, yielding \(2x^{-3}\). Notice how the negative sign flipped when multiplying the exponent.
The power rule makes calculating derivatives straightforward, allowing us to differentiate each term individually.
Simplifying Expressions
Simplifying expressions is an essential step before taking derivatives, especially when dealing with complex fractions or polynomials. It often makes it easier to apply rules like the power rule.Start off by simplifying the given function:
  • Take \( p(x) = \frac{2x^4 + 3x^2 - 1}{x^2} \).
  • Divide each term in the numerator by \( x^2 \).
  • You'll end up with \( 2x^2 + 3 - x^{-2} \).
Simplifying helps by breaking down more complicated expressions into basic polynomial terms. This step is crucial to ensure each term can then have the power rule applied, streamlining the process of finding derivatives and reducing potential mistakes.
First Derivative
The first derivative of a function gives the rate of change of the function concerning its variable, often denoting the slope or tangent at a point.Let's derive the first derivative for a function already simplified in terms of basic powers and constants, such as:
  • For \( 2x^2 \), the derivative is \( 4x \), indicating change proportional to \( x \).
  • For a constant like 3, the derivative is 0, showing no change.
  • For \(-x^{-2}\), you get \( 2x^{-3} \), reflecting rapid change due to the negative exponent.
After differentiating each component separately, combine them:
  • \( p'(x) = 4x + 0 - 2x^{-3} \)
  • The expression simplifies to \( p'(x) = 4x - 2x^{-3} \)
The first derivative is a powerful tool in calculus that helps understand how a function behaves. Whether the function increases, remains constant, or decreases can be clearly seen from \( p'(x) \).

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