A derivative is a mathematical concept that represents the rate of change of a function at a specific point. In other words, it measures how much a function is changing at a given point.
The first derivative of a function is another function that represents the rate of change of the original function. It is denoted as f'(x) or dy/dx and can be interpreted as the slope of the tangent line at a specific point on the original function.
The first derivative of a function can be calculated using the limit definition of a derivative, which is the change in the function's value divided by the change in the independent variable as the change approaches zero. It can also be calculated using differentiation rules such as the power rule, product rule, or quotient rule.
The second derivative of a function is the derivative of the first derivative. It represents the rate of change of the rate of change of the original function. It is denoted as f''(x) or d²y/dx² and can be interpreted as the concavity of the original function at a specific point.
The second derivative of a function can be calculated by taking the derivative of the first derivative. In other words, it is the derivative of the derivative. It can also be calculated using differentiation rules such as the power rule, product rule, or quotient rule.