Rules Of Derivatives Used in Economics ECO 401

Rules Of Derivatives Used in Economics (ECO 401) Constant Function rule: The derivative of a constant with respect to an independent variable x is zero. Let y = f (x) = C be a constant function, then y’ = f ’(x) = 0.

Examples: Find the derivatives of the following functions 1) f(x) = -30 f’(x) = ? 0 2) f(x) = f’(x) = ? 0 3) f(x) = 10 f’(x) = ? 0

The Power Function Rule A function of the form f (x) = xn is called a power function. This includes f (x) = x (where n = 1). The derivative of a power function, y=axn , is equal to the power n times the co-efficient a, multiplied by the variable x raised to power n-1. It can be expressed as,

Example Differentiate f (x) = x 5 Solution: By the power rule, the derivative of xn is n xn– 1. In our case n = 5, so we get f ’(x) = 5 x 5 -1 = 5 x 4

Examples: Find the derivatives of the following functions 1) f(x) = x 5 f’(x) = ? 5 x 4 2) f(x) = t-2 f’(x) = -2 t ? -3= 3) f(x) = t 3/2 f’(x) = ? 4) f(x) = f’(x) = 5) f(x) = = x-1 ? = u 2/3 f’(x) =? 6) f(x) = f’(x) =? = x-1/2

The Sums and Difference Rule If a function has terms linked by plus(or minus) signs, then the derivative of the function is equal to the Sum(or difference) of derivatives of the individual terms. y = f (x) = f(x) ± g(x), then y’ = f ’(x) = f’(x) ± g’(x)

Example Differentiate f (x) = 2 x 5 + x 4 – 4 x 3 + 8 x 2 – 3 x + 6. Solution: Apply the sum and difference rules, as well as the constant multiple property and the power rule. f ’(x) = 10 x 4 + 4 x 3 – 12 x 2 + 16 x -3

Examples: Find the derivatives of the following functions 1) f(x) = 4 x 4 -5 x 3+x 2 -3 x+7 f’(x) = ? 3) f(x) = f’(x) =? 2) f(x) = 10 - 7 x f’(x) = ? -2