Definition: Derivative (from Latin derivare, "to make less" or "to reduce") refers to any number which has a derivative in the denominator of any power. This concept can be used for both algebraic expressions and transcendental functions. In terms of algebraic expressions, derivatives are defined as the ratio of the increase/decrease in value of an expression with respect to its input (the independent variable) over an interval. For example, if we have a function \(f(x)\), then its derivative is defined as: \[ f'(x) = \frac{df}{dx} \] In terms of transcendental functions, derivatives are defined as the ratio of two differentials. The first differential (the one with the variable on the left hand side) is the derivative of the original function and is denoted by \(d/dx\) or \(f'(x)\), while the second differential (the one with the independent variable on the right hand side) is called the primitive of the original function, denoted by \(\frac{df}{dx}\). For instance, if we have a derivative of the function: \[ g(x) = ax^2 + b \] with \(a\) and \(b\) constants, its derivative with respect to \(x\) is given by: \[ g'(x) = 2ax + b \] In general, derivatives are also used in calculus and analysis to study the behavior of functions. They can be useful for solving problems involving rates of change or finding extrema. The word 'derivative' comes from Latin (derivare, "to make less" or "to reduce"), which has its roots in the meaning of "to remove." It's an adjective that means "of a derivative." For example, derivative is the inverse operation of differential.
closed bracket.
what should be said.