Differentiate the function y = 1/x.

The derivative of y = 1/x is -1/x^2.

To differentiate y = 1/x, we can use the power rule of differentiation. Let u = x, then y = u^(-1). Using the power rule, we have:

dy/du = -u^(-2)

Substituting back in x for u, we get:

dy/dx = -x^(-2)

Simplifying, we get:

dy/dx = -1/x^2

Therefore, the derivative of y = 1/x is -1/x^2. This means that the slope of the tangent line to the graph of y = 1/x at any point (x,y) is -1/x^2. We can also see that the derivative is negative for all x, which means that the function is decreasing for all x. Additionally, the derivative approaches zero as x approaches infinity or negative infinity, which means that the function approaches zero but never reaches it.

For those new to calculus, understanding the power rule can provide a foundation for this type of problem. More complex derivatives, such as those involving trigonometric functions, can be explored in the study of differentiation of trigonometric functions. To learn more about the techniques and rules of differentiation, consider reviewing basic differentiation rules. For a comprehensive introduction to the principles of derivatives, see Introduction to Derivatives.