Differentiate the following
$y=\frac{1}{x}$
Answer (1)
Rewrite the function: y = x 1 = x − 1 .
Apply the power rule: d x d y = − 1 ⋅ x − 2 .
Simplify the expression: d x d y = − x 2 1 .
The derivative of y = x 1 is − x 2 1 .
Explanation
Problem Analysis We are given the function y = x 1 and asked to find its derivative with respect to x .
Rewriting the Function To differentiate y = x 1 , we can rewrite it as y = x − 1 . This allows us to apply the power rule for differentiation, which states that if y = x n , then d x d y = n x n − 1 .
Applying the Power Rule Applying the power rule to y = x − 1 , we have
d x d y = ( − 1 ) x − 1 − 1 = − 1 x − 2 = − x − 2
Rewriting the Derivative Finally, we rewrite the derivative in fractional form:
d x d y = − x − 2 = − x 2 1
Examples
In physics, if x represents time and y = x 1 represents a quantity that decreases over time, the derivative d x d y = − x 2 1 tells us the rate at which that quantity is changing at any given time. For example, it could describe the rate of decay of a radioactive substance or the rate at which the intensity of light decreases with distance.
