Calculus – Derivatives

Definition of Derivative:

The derivative of a function f(x) at a point x = a is the instantaneous rate of change of the function at that point. Geometrically, it represents the slope of the tangent line to the graph of f(x) at x = a.

The following formulas give the Definition of Derivative. Scroll down the page for more examples and solutions.

 

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Interpretation of the Derivative as the Slope of a Tangent

The tangent line to y = f(x) at (a,f(a)) is the line through (a, f(a)) whose slope is equal to f’(a), the derivative of f at a. This means that the derivative is the slope of a curve at a given point on the curve.

Example:
Use the derivative to find the slope at any point along the following curves.
a) f(x) = 2x²
b) f(x) = \(\frac{2}{x}\)

Solution:
a) f(x) = 2x²

b)

 

Derivative Notations

If we use the traditional notation y = f(x) to indicate that the independent variable is x and the dependent variable is y, then some common notations for the derivatives are as follows:

 

What is a derivative?
Understanding the Definition of the Derivative

The following video shows how to find the slope of a tangent line to a curve and gives the definition of a derivative.

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The following video shows how to use the derivative to find the slope at any point along f(x) = x²

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Applications of Derivatives

1. Slope of a Curve:
   The derivative gives the slope of the tangent line to a curve at a specific point.
   
2. Velocity and Acceleration:
   If s(t) represents the position of an object at time t, then:
   Velocity: v(t)=s′(t)
   Acceleration: a(t)=v′(t)=s′′(t)
   
3. Optimization:
   Derivatives are used to find maximum or minimum values of functions (e.g., maximizing profit or minimizing cost).
   

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