Methods of Differentiation

Differentiation in BBA mathematics involves finding the derivative of a function, which represents how the function changes with respect to its variable. Below are the key methods of differentiation:

1. Basic Differentiation Rules

• These include applying standard rules such as the constant rule, power rule, and constant multiple rule:
  • Constant Rule: The derivative of a constant is zero.
  • Power Rule: ( \frac{d}{dx}(x^n) = n \cdot x^{n-1} ).
  • Constant Multiple Rule: ( \frac{d}{dx}(c \cdot f(x)) = c \cdot f'(x) ), where ( c ) is a constant.

2. Sum and Difference Rule

• When differentiating a function that is the sum or difference of two or more functions, differentiate each term separately.
• Formula: [ \frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x) ]

3. Product Rule

• Used when differentiating the product of two functions.
• Formula: [ \frac{d}{dx}(f(x) \cdot g(x)) = f(x) \cdot g'(x) + g(x) \cdot f'(x) ]
• This ensures that both functions are accounted for in the derivative.

4. Quotient Rule

• Applied when differentiating a quotient of two functions.
• Formula: [ \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{[g(x)]^2} ]
• This is useful for differentiating fractions where one function is divided by another.

5. Chain Rule

• Used for finding the derivative of a composite function (a function within another function).
• Formula: [ \frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x) ]
• The chain rule is helpful for differentiating functions with nested expressions.

6. Implicit Differentiation

• When a function is not explicitly solved for one variable in terms of another (e.g., (x) and (y) are mixed), implicit differentiation is used.
• Differentiate both sides of the equation with respect to (x), applying the chain rule as needed.
• Solve for ( \frac{dy}{dx} ).

7. Logarithmic Differentiation

• This method involves taking the natural logarithm of both sides of a function before differentiating.
• Useful for differentiating functions where the variable appears as both the base and exponent (e.g., ( y = x^x )).
• Steps:
  1. Take the natural log of both sides: ( \ln(y) = \ln(f(x)) ).
  2. Differentiate both sides using the chain rule: ( \frac{1}{y} \frac{dy}{dx} = f'(x) ).
  3. Solve for ( \frac{dy}{dx} ).

8. Higher-Order Derivatives

• These involve differentiating a function multiple times.
• The second derivative, ( f''(x) ), represents the derivative of the first derivative and measures the rate of change of the slope.
• Higher-order derivatives can be found by repeatedly applying differentiation.

9. Differentiation of Exponential and Logarithmic Functions

• For exponential functions ( f(x) = e^x ), the derivative is ( f'(x) = e^x ).
• For logarithmic functions ( f(x) = \ln(x) ), the derivative is ( f'(x) = \frac{1}{x} ).

10. Differentiation of Trigonometric Functions

• Common derivatives for trigonometric functions include:
  • ( \frac{d}{dx}(\sin(x)) = \cos(x) )
  • ( \frac{d}{dx}(\cos(x)) = -\sin(x) )
  • ( \frac{d}{dx}(\tan(x)) = \sec^2(x) )

These methods provide a range of techniques for solving problems related to rates of change, optimization, and other calculus-based applications in business administration.