Calculating the Derivative

Derivative Notations

The derivative of a function measures how the function changes as its input changes. Several notations are used to represent the derivative of a function y = f(x):

• f′(x)

• y′

• \frac{dy}{dx}

• \frac{df}{dx}

• \frac{d}{dx}[f(x)]

• D_x[f(x)]

Each notation expresses the rate of change of y with respect to x.

Rules for Derivatives

• Derivative of a Constant: If f(x) = k, then f′(x) = 0.

• Simple Power Rule: If y = x^n, then

• Derivative of y = x:

• Derivative of a Constant Times a Function:

• Derivative of y = c x:

• Derivative of a Sum or Difference:

Example: For f(x) = 6x^4 + 2x^3 - 2x^2 + 4x + 6, the derivative is:

Marginal Analysis

Economic Applications of Derivatives

Derivatives are used in business to analyze changes in revenue, cost, and profit:

• Marginal Revenue (MR): The derivative of the revenue function R(x).

• Marginal Cost (MC): The derivative of the cost function C(x).

• Marginal Profit (MP): The derivative of the profit function P(x).

Example: If C(x) = 0.15x^2 - 18x + 960, then gives the marginal cost.

Derivatives of Products and Quotients

Product Rule

To differentiate the product of two functions:

• If f(x) = u(x) v(x), then

Example: For f(x) = (x^2 - 1)(3x + 1):

Quotient Rule

To differentiate the quotient of two functions:

• If f(x) = \frac{u(x)}{v(x)}, then

Example: For f(x) = \frac{2x - 1}{x + 2}:

The Chain Rule

Composition of Functions

Functions can be combined by composition, where the output of one function becomes the input of another. The composite function f ∘ g is defined as (f ∘ g)(x) = f(g(x)).

• Domain: The domain of f ∘ g is all x in the domain of g such that g(x) is in the domain of f.

Example: If f(x) = 6 - x and g(x) = x^3 + 4, then (f ∘ g)(x) = 6 - (x^3 + 4).

The Chain Rule

The Chain Rule is used to differentiate composite functions. If y = f(u) and u = g(x), then y = f(g(x)), and:

• Alternatively,

General Power Rule: If y = [g(x)]^n, then

Derivatives of Exponential Functions

Natural Exponential Function

The derivative of the natural exponential function is:

Chain Rule for Exponential Functions:

General Exponential Function:

Derivatives of Logarithm Functions

Logarithmic Functions

The derivative of the logarithm function is:

Natural Logarithm:

Chain Rule for Logarithmic Functions:

Example: For f(x) = 6x^2 \ln x,

Summary Table: Derivative Rules