Alternate Definition of Derivative
limit (as x approaches a number c)=f(x)-f(c)/x-c x≠c
An alternate definition of the derivative of a function f(x) is in terms of the difference quotient, which measures the rate at which the function changes as the input x changes. The difference quotient is defined as:
f'(x) = lim(h->0) [f(x + h) – f(x)]/h
where h is a small change in x, and the limit is taken as h approaches zero.
This definition states that the derivative of a function at a point x is the limit of the difference quotient as h approaches zero. Geometrically, the derivative represents the slope of the tangent line to the graph of the function at the point (x, f(x)).
The difference quotient can be thought of as an approximation to the instantaneous rate of change of the function at x. As h gets smaller, the approximation gets better, and the limit gives the exact value of the derivative.
This alternate definition of the derivative is closely related to the limit definition, as they both describe the same concept in different ways. The difference quotient definition is often used to compute derivatives of simple functions by algebraic manipulation, while the limit definition is more commonly used to prove general properties of derivatives and to compute derivatives of more complicated functions using calculus techniques such as the chain rule and product rule.
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