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Chapter 2: Problem 39

Find \(f^{\prime}(x)\). $$f(x)=\frac{1}{\sqrt{x}}$$

Short Answer

Expert verified
\[f^{\prime}(x) = -\frac{1}{2}x^{-\frac{3}{2}}\]

Step by step solution

01

- Rewrite the Function

Rewrite the function in a form that makes it easier to differentiate. Instead of writing it as a fraction, use a negative exponent: \[ f(x) = \frac{1}{\sqrt{x}} = x^{-\frac{1}{2}} \]
02

- Apply the Power Rule

Use the power rule of differentiation, which states that if \( f(x) = x^n \), then \( f^{\prime}(x) = nx^{n-1} \). Here, our \( n \) is \( -\frac{1}{2} \): \[ f^{\prime}(x) = -\frac{1}{2}x^{-\frac{1}{2}-1} \]
03

- Simplify the Expression

Simplify the derivative: \[ f^{\prime}(x) = -\frac{1}{2}x^{-\frac{3}{2}} \] This can also be written as: \[ f^{\prime}(x) = -\frac{1}{2}\frac{1}{x^{\frac{3}{2}}} \]

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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 for finding the derivative of a function of the form \( f(x) = x^n\). This rule states that if \( f(x) = x^n\), then the derivative of the function, \( f^{\textprime}(x)\), is given by \( nx^{n-1}\). To apply the Power Rule, you simply:
  • Take the exponent \( n\) and multiply it by the coefficient of \( x\).
  • Subtract 1 from the exponent \( n\).
For example, if \( f(x) = x^3\), the derivative would be \( f^{\textprime}(x) = 3x^2\). This method can be applied to any polynomial, negative exponent, or fractional exponents. Understanding and practicing the Power Rule is essential for mastering differentiation in calculus.
Negative Exponents
Negative exponents are a way of expressing reciprocals. For instance, \( x^{-n}\) is equivalent to \( \frac{1}{x^n}\). When differentiating functions with negative exponents, the Power Rule still applies smoothly.
Let's look at an example where a negative exponent is used in differentiation:
Suppose you have \( f(x) = x^{-3}\), then using the Power Rule, \( f^{\textprime}(x) = -3x^{-4}\).
Steps to handle negative exponents include:
  • Rewrite the function without denominators.
  • Apply the Power Rule normally.
  • Simplify the result if possible.
This method holds even if the exponent is a fraction and negative. It鈥檚 important to rewrite expressions to a form where these rules can be easily applied.
Simplifying Derivatives
Simplifying derivatives is a crucial step to make the final answer clear and concise. In many cases, after applying the Power Rule, you'll need to simplify expressions, especially those containing negative or fractional exponents.
Let's revisit the example provided:
  • First, rewrite \( f(x) = \frac{1}{\sqrt{x}}\) as \( x^{-\frac{1}{2}}\).
  • Differentiate using the Power Rule to get \( f^{\textprime}(x) = -\frac{1}{2}x^{-\frac{3}{2}}\).
  • Finally, express the derivative in simplest form: \( f^{\textprime}(x) = -\frac{1}{2}\cdot \frac{1}{x^{\frac{3}{2}}}\).
Simplifying derivatives helps in understanding the behavior and properties of functions. Always look for opportunities to combine like terms, reduce fractions, and convert negative exponents back into fractions when appropriate.

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