Derivatives are a fundamental tool of calculus. They are used to measure how a function changes over time and can be used to calculate everything from interest rates to predicting how things will act in the future. Here are some of the main uses of derivatives:

• Finding the rate of change of a quantity: Derivatives can be used to find the rate at which one quantity is changing with respect to another quantity.

• Finding the slope of a curve: The slope of a curve at a particular point is equal to the derivative of the function at that point.

• Finding maxima and minima of a function: Derivatives can be used to find the points where a function reaches its maximum or minimum value.

• Finding the equation of a tangent line to a curve: The derivative can be used to find the equation of a tangent line to a curve at a particular point.

• Approximation: Derivatives can be used to approximate the value of a function at a particular point.

• Optimization: Derivatives are used in optimization algorithms to find the maximum or minimum value of a function.

• Financial contracts: Derivatives are also used in finance as a type of financial contract whose value is dependent on an underlying asset, group of assets, or benchmark.

Overall, derivatives have wide usage in mathematics, science, engineering, physics, economics, and finance.