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The derivative of a function measures the instantaneous rate of change of the function. It is a fundamental concept in calculus and has applications in many fields, including physics, engineering, economics, and biology.

Definition

The derivative of a function $ f(x) $ at a point $ x=a $ , denoted by $ f’(a) $ , is defined as the limit:

$$ f’(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} $$ provided the limit exists.

Interpretation of the Derivative

• Slope of the Tangent Line: The derivative $ f’(a) $ represents the slope of the tangent line to the graph of $ f(x) $ at the point $ (a, f(a)) $ .

• Instantaneous Rate of Change: The derivative $ f’(a) $ represents the instantaneous rate of change of $ f(x) $ at $ x=a $ .

Derivative Rules

Applications of Derivatives

Derivatives have numerous applications in calculus and beyond. Some key applications include:

• Finding Critical Points: Critical points are points where the derivative is zero or undefined. These points can indicate local maxima, minima, or points of inflection.
• Optimization: Derivatives can be used to find the maximum or minimum values of a function.
• Related Rates: Derivatives can be used to find the rate of change of one variable with respect to another.
• Local Linearity: The derivative can be used to approximate the value of a function near a given point.

Higher Order Derivatives

The derivative of a function can be differentiated again to find the second derivative, denoted by $ f’’(x) $ . This process can be repeated to find the third derivative, fourth derivative, and so on. Higher order derivatives are used to study the concavity of a function and its points of inflection.

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function that is defined implicitly by an equation. For example, the equation $ x^2 + y^2 = 1 $ defines a circle. To find the derivative of $ y $ with respect to $ x $ , we differentiate both sides of the equation implicitly.

Trigonometric Derivatives

Examples

• Example 1: Find the derivative of $ f(x) = x^3 + 2x $ .

  Using the power rule and the sum rule, we get:

  $$ f’(x) = 3x^2 + 2 $$

• Example 2: Find the derivative of $ g(x) = \sin(x^2) $ . Using the chain rule, we get: $$ g’(x) = \cos(x^2) \cdot 2x = 2x \cos(x^2) $$