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Derivative of a Fraction Using the Power Rule

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Derivative of a Fraction Using the Power Rule

In calculus, the power rule is a fundamental tool for finding the derivative of a function. This rule states that the derivative of a term of the form xn is nx(n-1). While this rule is straightforward for simple terms, it can become more complex when dealing with fractions. This blog post will guide you through the process of finding the derivative of a fraction using the power rule.

Understanding the Power Rule

Before we delve into fractions, let's revisit the power rule itself. The power rule states that for any real number n, the derivative of xn is given by:

d/dx (xn) = nx(n-1)

For example, the derivative of x3 is 3x2. This rule applies to both positive and negative values of n. For instance, the derivative of x-2 is -2x-3.

Applying the Power Rule to Fractions

When dealing with fractions, we can rewrite them as terms with negative exponents. For example, 1/x can be written as x-1. This allows us to apply the power rule directly.

Example 1: Finding the Derivative of 1/x

Let's find the derivative of 1/x:

  1. Rewrite 1/x as x-1.
  2. Apply the power rule: d/dx (x-1) = -1x-2.
  3. Simplify: -1x-2 = -1/x2.

Therefore, the derivative of 1/x is -1/x2.

Example 2: Finding the Derivative of 1/x2

Let's find the derivative of 1/x2:

  1. Rewrite 1/x2 as x-2.
  2. Apply the power rule: d/dx (x-2) = -2x-3.
  3. Simplify: -2x-3 = -2/x3.

Therefore, the derivative of 1/x2 is -2/x3.

Key Points to Remember

  • Rewrite fractions as terms with negative exponents to apply the power rule.
  • Remember that the power rule involves multiplying the coefficient by the power and then reducing the power by 1.
  • Simplify the derivative expression after applying the power rule.

Conclusion

Finding the derivative of a fraction using the power rule is a straightforward process once you understand the concept of rewriting fractions with negative exponents. By applying the power rule and simplifying the result, you can successfully find the derivative of any fraction. This knowledge is essential for understanding various calculus concepts and their applications.

fractions