Question

Is the standard expression defining the derivative of some function

Answer 1

The standard expression defining the derivative of a function is f'(x) = lim(h -> 0) [f(x + h) - f(x)] / h.

Step-by-step explanation:

The derivative of a function represents the rate at which the function's output changes concerning its input. The standard expression for the derivative, denoted as f'(x), is defined as the limit of the difference quotient as the interval h approaches zero. In mathematical terms:

`\[ f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h} \]`

This expression captures the instantaneous rate of change of the function at a specific point x. Breaking it down,
`\( f(x + h) - f(x) \)`represents the change in the function's output over the interval h, and dividing by h ensures that the rate of change is calculated as h approaches zero, essentially providing the slope of the tangent line at the point x.

In essence, this expression encapsulates the fundamental concept of calculus, emphasizing the infinitesimally small changes in the function's input and output to determine the slope of the tangent line at a given point. It is a foundational tool in understanding and analyzing the behavior of functions, enabling mathematicians and scientists to model and comprehend dynamic systems and phenomena.