Question

If a function has a constant derivative then it is linear, and conversely.

Short answer

A function is linear if its derivative is a constant, and conversely if a function has a constant derivative, it is linear. As such, the derivative can be used to determine whether a function is linear or not.

Step-by-step solution

Understanding Linearity

A function is linear if it can be defined by \(y = mx + b\), where m and b are constants. In other words, every increase in x results in a proportional increase in y.

Understanding the Derivative

The derivative of a function measures the rate at which the function changes for a given change in input. The derivative of a constant function is zero, but for any other function the derivative is a measure of how much the slope of the function changes for small changes in input.

Relating Linearity and Derivatives

When the derivative of a function is a constant, it means that its rate of change is constant across all inputs. Therefore, for a constant derivative, the function is linear. Similarly, for a linear function, the derivative is a constant.