What is the derivative of y = 2/x?

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

To find the derivative of \( y = \frac{2}{x} \), we can use the rules of differentiation. First, it's helpful to rewrite the function in a form that makes it easier to differentiate. We can express \( \frac{2}{x} \) as \( 2x^{-1} \). This is because dividing by \( x \) is the same as multiplying by \( x \) raised to the power of \(-1\).

Now, we apply the power rule for differentiation. The power rule states that if you have a function of the form \( ax^n \), where \( a \) is a constant and \( n \) is a power, the derivative is \( anx^{n-1} \). In our case, \( a = 2 \) and \( n = -1 \).

Using the power rule:
\[ \frac{d}{dx}(2x^{-1}) = 2 \cdot (-1) \cdot x^{-1-1} \]
\[ = -2x^{-2} \]

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

This means that for any value of \( x \), the rate of change of \( y \) with respect to \( x \) is given by \( -\frac{2}{x^2} \). This negative sign indicates that as \( x \) increases, \( y \) decreases, and the magnitude of the rate of change depends on the square of \( x \).