Alternative form of the definition of the derivative
lim x->c. f(x)-f(c) / x-c
The derivative of a function f at a point x is defined as the limit of the difference quotient as the interval h approaches zero:
f'(x) = lim(h->0) (f(x+h) – f(x))/h
This definition gives the instantaneous rate of change of a function at any point x in its domain, which can also be interpreted as the slope of the tangent line to the graph of the function at that point.
Alternatively, the derivative can also be defined as the function that assigns to each value of x the slope of the tangent line to the graph of f at that point:
f'(x) = dy/dx
This definition emphasizes the fact that the derivative is a function itself, rather than a limit or a ratio, and can be used to calculate the slope of the tangent line at any point x. Both definitions are equivalent and provide a powerful tool for analyzing the behavior of functions in calculus.
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