SHORTCUTS TO DERIVATIVES u and v are functions (x + 8) c is a constant (4) 1.f(x) = cf’(x) = 0 f(x) = 6

2. f(x) = x n f’(x) = nx n-1 f(x) = x 4

3. f(x) = cuf’(x) = c(du/dx) f(x) = 2x 4

4.f(x) = u + vf’(x) = (du/dx) + (dv/dx) f(x) = 2x 2 + x

5. f(x) = uvf’(x) = u(dv/dx) + v(du/dx) f(x) = (x)(2x 2 )

6. f(x) = u/vf’(x) = (v(du/dx) – u(dv/dx))/v 2 f(x) = 2x 2 /(x+1)

HIGHER ORDER DERIVATIVE Most of the time, the derivative will be another function (that is why we write as f’(x)) We can take the derivative on ANY function so we can take the derivative of a derivative. The derivative of a derivative is called the second derivative: f’’(x) or y’’ or d 2 y/dx 2 The derivative of a a second derivative: The third derivative f’’’(x) = y’’’

f(x) = x 3 – 5x f’(x) = f’’(x) = f’’’(x) = f’’’’(x) =

POWER RULE - REVISITED f(x) = x n f’(x) = nx n-1 What happens if I have a function raised to a power? f(x) = (3x + 2) 2 f’(x) = f(x) = 9x x + 4f’(x) = 18x + 12 We are going to change the rule to make it apply to other power situations f(x) = u n f’(x) = nu n-1 (du/dx) f’’(x) = 2(3x + 2) 1 (3) = 6(3x + 2) = 18x + 12