A first derivative, also known as the derivative or differential coefficient, is a mathematical concept in calculus that represents the rate of change of a function at a specific point. It is the slope of the tangent line to the function at that point.
To find the first derivative of a function, you can use the power rule, product rule, quotient rule, or chain rule, depending on the form of the function. These rules involve taking the derivative of each term in the function and combining them using algebraic operations.
The first derivative has many applications in science and engineering, as it represents the instantaneous rate of change of a physical quantity. It can be used to find maximum and minimum values, acceleration, and other important properties of a function.
The first derivative can be interpreted graphically as the slope of the tangent line at each point on the graph of a function. A positive first derivative indicates a function is increasing, while a negative first derivative indicates a function is decreasing. A first derivative of zero indicates a point of inflection or a horizontal tangent line.
The second derivative is the derivative of the first derivative. It represents the rate of change of the first derivative, which is the rate of change of the original function. The second derivative can provide additional information about the behavior of a function, such as the concavity and points of inflection.