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Definitions

Derivatives are a fundamental concept in calculus, representing the rate at which a function is changing at any given point. The formal definition and various notations are as follows:

Formal Definition​

For a function y=f(x)y = f(x), the derivative of ff at a point xx is given by the limit:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{{h \to 0}} \frac{f(x + h) - f(x)}{h}

Equivalent Notations for Derivatives​

There are several notations used interchangeably to denote the derivative of ff with respect to xx:

  • f′(x)f'(x)
  • y′y'
  • dydx\frac{dy}{dx}
  • ddxf(x)\frac{d}{dx} f(x)
  • Df(x)Df(x)

Notation for Derivatives at a Specific Point​

When evaluating the derivative at a specific point x=ax = a, the notations adapt slightly:

f′(a)=dydx∣x=a=ddxf(x)∣x=a=Df(a)f'(a) = \left. \frac{dy}{dx} \right|_{x=a} = \left. \frac{d}{dx} f(x) \right|_{x=a} = Df(a)

The derivative f′(a)f'(a) represents the slope of the tangent line to the graph of ff at the point aa. It can also be interpreted as the instantaneous rate of change of the function f(x)f(x) at x=ax = a.

Interpretation of the Derivative​

The derivative of a function provides several critical insights into the behavior of the function. Here are the main interpretations:

Slope of the Tangent Line​

  • Slope (m): For y=f(x)y = f(x), the derivative f′(a)f'(a) at a point aa represents the slope mm of the tangent line to the curve y=f(x)y = f(x) at that point.

Equation of the Tangent Line​

  • Tangent Line Equation: Given a function y=f(x)y = f(x) and a point x=ax = a, the equation of the tangent line to the curve at that point is:
y=f(a)+f′(a)(x−a)y = f(a) + f'(a)(x - a)

Rate of Change​

  • Instantaneous Rate of Change: The derivative f′(a)f'(a) indicates the instantaneous rate of change of the function f(x)f(x) at the point x=ax = a.

Physical Interpretation in Motion​

  • Velocity: If f(t)f(t) represents the position of an object at time tt, then f′(a)f'(a) corresponds to the velocity of the object at time t=at = a.

References​