How to find the derivative 1/x

To find the derivative of the function $f(x) = \frac{1}{x}$, we will use basic differentiation rules. Here are the steps:

Step 1: Recognize the Function Form

The function $\frac{1}{x}$ can be rewritten using a negative exponent:

Step 2: Apply the Power Rule

The power rule for differentiation states that if $f(x) = x^n$, then $f'(x) = n x^{n-1}$. For this function, $n = -1$:

Step 3: Simplify the Expression

Now, simplify the expression:

This can be written back in fractional form:

Conclusion

So, the derivative of $\frac{1}{x}$ is:

Important Note

Memorizing this result can be very useful as it frequently appears in calculus problems. Basically:

Understanding this differentiation helps in solving complex calculus problems more efficiently.

Understanding the Power Rule in Calculus

The power rule is a fundamental concept in differential calculus, used to find the derivative of functions of the form $f(x) = x^n$. This rule simplifies the differentiation process, making it essential for calculus students and professionals.

The Power Rule Formula

For a function $f(x) = x^n$, where $n$ is any real number, the derivative $f'(x)$ is given by:

How It Works

Step-by-step:

1. Identify the exponent $n$.
2. Multiply the function by the exponent $n$.
3. Subtract 1 from the exponent.

Examples

1. Example 1: $f(x) = x^3$

   • Applying the power rule:
   $$f'(x) = 3x^{3-1} = 3x^2$$

2. Example 2: $f(x) = x^{-2}$

   • Applying the power rule:
   $$f'(x) = -2x^{-2-1} = -2x^{-3}$$

3. Example 3: $f(x) = x^{\frac{1}{2}}$

   • Applying the power rule:
   $$f'(x) = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}}$$

Important Points

• The power rule applies directly to any real number exponent.
• It does not require complex calculations or advanced techniques.
• This makes the power rule one of the most efficient tools for differentiating polynomial functions.

Understanding and mastering the power rule is crucial, as it lays the foundation for more advanced topics in calculus.

Understanding Negative Exponent

A negative exponent indicates that the base should be reciprocated. In mathematical terms, a negative exponent shifts the base to the denominator of a fraction and converts the exponent to a positive value.

Basic Concept

For a nonzero number $a$ and a positive integer $n$:

$a^{-n} = \frac{1}{a^n}$

This is a crucial rule to remember. It tells us that we simply take the reciprocal of the base and then raise it to the positive exponent.

Example

Let's see how this works with an example:

$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

General Form

You can generalize this concept for any real number and exponent as:

$a^{-x} = \frac{1}{a^x}$

where $a \neq 0$.

Key Points

• If the base is positive and the exponent is negative, the result will be a positive fraction.
• If the base is negative and the exponent is even, the result will be a positive fraction.
• If the base is negative and the exponent is odd, the result will be a negative fraction.

Understanding negative exponents is fundamental in algebra and higher-level mathematics, as they help simplify expressions and solve equations that involve powers and roots. Always remember: flipping the sign of the exponent means flipping the base to the other side of the fraction.