What is the Derivative of 1/x?

Differentiating a basic function like 1/x is a vital step toward mastering more complex calculations. In this article, we’ll break down the process in a straightforward way to help you find the derivative of 1/x easily.

To find the derivative of a function, we’re essentially figuring out how it changes when we tweak its independent variable.

In the case of 1/x, our independent variable is ‘x,’ and we want to find the derivative, which shows us how 1/x changes concerning ‘x.’

The derivative of 1/x, written as d(1/x)/dx or dy/dx, can be found using the power rule for differentiation. The power rule says that if you have a function in the form of xⁿ, the derivative is

nx^(n-1). In our case, ‘n’ is -1 because we can rewrite 1/x as x^(-1).

Applying the power rule:

dy/dx = -1 x^(-1-1) = -1 . x^(-2) = -1/x²

So, the result of differentiating 1/x is -1/x².

Now, let’s make sense of our solution. The derivative, -1/x², tells us how the function 1/x behaves as ‘x’ changes.

1. Negative Sign: The minus sign indicates that as ‘x’ increases, 1/x decreases, and as ‘x’ decreases, 1/x increases. In simpler terms, the function goes down when ‘x’ goes up, and it goes up when ‘x’ goes down.
2. x² in the Denominator: The x² in the denominator means that the rate of change of 1/x is inversely related to the square of ‘x.’ When ‘x’ gets larger, the rate of change decreases quite rapidly, and when ‘x’ gets smaller, the rate of change increases significantly.