University:
York University
Course:
MATH 1013 | Applied Calculus I
Academic year:

2022
Views:

91

Pages:

4
Author:

Nick T.

Chapter 3: Differentiation Rules Derivatives of Polynomials & Exponential Functions Constant Functions f(x) = c (horizontal line) Slope = 0 --> f'(x) = 0 f'(x) = lim(h->0) [f(x+h) - f(x)] / h = lim(h->0) 0 / h = 0 d/dx(c) = 0 Power Functions f(x) = x^n (n a positive integer) If n equals 1, then f'(x) = lim(h->0) [f(x+h) - f(x)] / h = lim(h->0) [(x+h)^4 - x^4] / h (x^3 + 3x^2h + 3xh^2 + h^3)(x+h) 4x^3 + 3x^2h + 3xh^2 + h^3 x^3h + 3x^2h^2 + 3xh^3 + h^4 4x^3 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4 4x^3h + 6x^2h^2 + 4xh^3 + h^4 h(4x^3 + 6x^2h + 4xh^2 + h^3) / h = 4x^3 d/dx(x) = 1 f(x) = x, y = x --> f'(x) = 1 Derivative of d/dx(x^2) = 2x d/dx(x^n) = nx^(n-1) d/dx(x^3) = 3x^2 Front lim(x->a) [f(x) - f(a)] / (x - a) f'(x) = lim(h->0) [f(x+h) - f(x)] / h = lim(h->0) [(x+h)^n - x^n] / h Power functions with negative exponents d/dx(1/x) = -1/x^2 d/dx(x^-1) = (-1)x^-2 Exponent in a Fraction Differentiation formula F-force m-first mass M-second mass r-distance between bodies G-gravitational constant F=GmM/r^2 df/dr = [(GmM)r^-2]' = (-2)(GmM)r^-3 = -2GmM/r^3 This is the rate at change of the force f(x) = 1/x^2 f'(x) = (-2)x^-3 = -2/x^3 y = x^(1/2) y' = (1/2)x^(-1/2) = 1/(2√x) f(x) = x^(2/3) f'(x) = (2/3)x^(-1/3) = 2/(3√x) y = 1 - 2(x - 1)