The basic rules of differentiation of functions in calculus are presented along with several examples.
1. Derivative of a Constant Function
The derivative of \( f(x) = c \), where \( c \) is a constant, is given by:
Example & Solution
Given \( f(x) = -10 \), hence:
\[ f'(x) = 0 \]2. Derivative of a Power Function (Power Rule)
The derivative of \( f(x) = x^r \), where \( r \) is a constant real number, is given by:
Example & Solution
Given \( f(x) = x^{-2} \), hence:
\[ f'(x) = -2x^{-2-1} = \dfrac{-2}{x^3} \]3. Derivative of a Function Multiplied by a Constant
The derivative of \( f(x) = c \cdot g(x) \), where \( c \) is a constant, is given by:
Example & Solution
Given \( f(x) = 3x^3 \), let \( c = 3 \) and \( g(x) = x^3 \), hence \( f(x) = c \cdot g(x) \):
\[ f'(x) = c \cdot g'(x) = 3(3x^2) = 9x^2 \]4. Derivative of the Sum of Functions (Sum Rule)
The derivative of \( f(x) = g(x) + h(x) \) is given by:
Example & Solution
Given \( f(x) = x^2 + 4 \), let \( g(x) = x^2 \) and \( h(x) = 4 \):
\[ f'(x) = g'(x) + h'(x) = 2x + 0 = 2x \]5. Derivative of the Difference of Functions (Difference Rule)
The derivative of \( f(x) = g(x) - h(x) \) is given by:
Example & Solution
Given \( f(x) = x^3 - x^{-2} \), let \( g(x) = x^3 \) and \( h(x) = x^{-2} \):
\[ f'(x) = g'(x) - h'(x) = 3x^2 - (-2x^{-3}) = 3x^2 + 2x^{-3} \]6. Derivative of the Product of Two Functions (Product Rule)
The derivative of \( f(x) = g(x) \cdot h(x) \) is given by:
Example & Solution
Given \( f(x) = (x^2 - 2x)(x - 2) \), let \( g(x) = x^2 - 2x \) and \( h(x) = x - 2 \):
\[ f'(x) = g(x)h'(x) + h(x)g'(x) = (x^2 - 2x)(1) + (x - 2)(2x - 2) \]Expand and group like terms:
\[ f'(x) = x^2 - 2x + 2x^2 - 6x + 4 = 3x^2 - 8x + 4 \]7. Derivative of the Quotient of Two Functions (Quotient Rule)
The derivative of \( f(x) = \dfrac{g(x)}{h(x)} \) is given by:
Example & Solution
Given \( f(x) = \dfrac{x - 2}{x + 1} \), let \( g(x) = x - 2 \) and \( h(x) = x + 1 \). Here \( g'(x) = 1 \) and \( h'(x) = 1 \):
\[ f'(x) = \dfrac{(x + 1)(1) - (x - 2)(1)}{(x + 1)^2} \]Group and simplify:
\[ f'(x) = \dfrac{x + 1 - x + 2}{(x + 1)^2} = \dfrac{3}{(x + 1)^2} \]