What is the derivative of y = 1/x?

The derivative of \( y = \frac{1}{x} \) is \( y' = -\frac{1}{x^2} \).

To understand why this is the case, let's break it down step by step. When we talk about the derivative, we are looking for the rate at which \( y \) changes with respect to \( x \). In other words, we want to find the slope of the tangent line to the curve at any point.

First, rewrite \( y = \frac{1}{x} \) in a form that is easier to differentiate. We can express it as \( y = x^{-1} \). This is because \( \frac{1}{x} \) is the same as \( x \) raised to the power of \(-1\).

Next, we use the power rule for differentiation. The power rule states that if you have a function \( y = x^n \), then its derivative \( y' \) is \( nx^{n-1} \). Applying this rule to our function \( y = x^{-1} \), we get:

\[ y' = -1 \cdot x^{-1-1} \]

Simplifying the exponent, we have:

\[ y' = -1 \cdot x^{-2} \]

Finally, we can rewrite \( x^{-2} \) as \( \frac{1}{x^2} \). So, the derivative of \( y = \frac{1}{x} \) is:

\[ y' = -\frac{1}{x^2} \]

This tells us that for any value of \( x \), the rate of change of \( y \) with respect to \( x \) is \(-\frac{1}{x^2} \). This negative sign indicates that as \( x \) increases, \( y \) decreases, and vice versa.