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Differentiation Rules

In BBA (Business Administration) mathematics, differentiation refers to finding the derivative of a function, which represents the rate at which the function is changing at any given point. Below are the fundamental rules of differentiation:

1. Constant Rule

  • The derivative of a constant is zero.
  • Formula: ( \frac{d}{dx}(c) = 0 ), where (c) is a constant.
  • Example: ( \frac{d}{dx}(5) = 0 ).

2. Power Rule

  • To differentiate a power of (x), multiply by the exponent and reduce the exponent by one.
  • Formula: ( \frac{d}{dx}(x^n) = n \cdot x^{n-1} ), where (n) is any real number.
  • Example: ( \frac{d}{dx}(x^3) = 3x^2 ).

3. Constant Multiple Rule

  • The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
  • Formula: ( \frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x)) ).
  • Example: ( \frac{d}{dx}(3x^2) = 3 \cdot 2x = 6x ).

4. Sum Rule

  • The derivative of a sum of functions is the sum of their derivatives.
  • Formula: ( \frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x)) ).
  • Example: ( \frac{d}{dx}(x^2 + 3x) = 2x + 3 ).

5. Difference Rule

  • The derivative of a difference of functions is the difference of their derivatives.
  • Formula: ( \frac{d}{dx}(f(x) - g(x)) = \frac{d}{dx}(f(x)) - \frac{d}{dx}(g(x)) ).
  • Example: ( \frac{d}{dx}(x^2 - 3x) = 2x - 3 ).

6. Product Rule

  • To differentiate the product of two functions, use: ( (f(x) \cdot g(x))' = f(x) \cdot g'(x) + g(x) \cdot f'(x) ).
  • Formula: ( \frac{d}{dx}(f(x) \cdot g(x)) = f(x) \cdot \frac{d}{dx}(g(x)) + g(x) \cdot \frac{d}{dx}(f(x)) ).
  • Example: If ( f(x) = x^2 ) and ( g(x) = 3x ), then ( \frac{d}{dx}(x^2 \cdot 3x) = x^2 \cdot 3 + 3x \cdot 2x = 9x^2 ).

7. Quotient Rule

  • To differentiate a quotient of two functions, use: ( \left(\frac{f(x)}{g(x)}\right)' = \frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{[g(x)]^2} ).
  • Formula: ( \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x) \cdot \frac{d}{dx}(f(x)) - f(x) \cdot \frac{d}{dx}(g(x))}{[g(x)]^2} ).
  • Example: If ( f(x) = x ) and ( g(x) = x + 1 ), then ( \frac{d}{dx}\left(\frac{x}{x+1}\right) = \frac{(x+1) \cdot 1 - x \cdot 1}{(x+1)^2} = \frac{1}{(x+1)^2} ).

8. Chain Rule

  • To differentiate a composite function, use the derivative of the outer function times the derivative of the inner function.
  • Formula: ( \frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x) ).
  • Example: If ( f(x) = (3x + 2)^2 ), then let ( g(x) = 3x + 2 ) and ( f(g) = g^2 ), giving ( \frac{d}{dx}(f(x)) = 2g \cdot 3 = 6(3x + 2) ).

These rules form the foundation for solving various problems involving rates of change, optimization, and other applications in calculus.

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