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SHORTCUTS TO DERIVATIVES

SHORTCUTS TO DERIVATIVES. u and v are functions (x + 8) c is a constant (4) f(x) = c f’(x) = 0 f(x) = 6. 2. f(x) = x n f’(x) = nx n-1 f(x) = x 4. 3. f(x) = cu f’(x) = c(du/dx) f(x) = 2x 4. f(x) = u + v f’(x) = (du/dx) + (dv/dx) f(x) = 2x 2 + x.

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SHORTCUTS TO DERIVATIVES

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

  2. 2. f(x) = xn f’(x) = nxn-1 • f(x) = x4

  3. 3. f(x) = cu f’(x) = c(du/dx) • f(x) = 2x4

  4. f(x) = u + v f’(x) = (du/dx) + (dv/dx) f(x) = 2x2 + x

  5. 5. f(x) = uv f’(x) = u(dv/dx) + v(du/dx) • f(x) = (x)(2x2)

  6. 6. f(x) = u/v f’(x) = (v(du/dx) – u(dv/dx))/v2 • f(x) = 2x2/(x+1)

  7. 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 d2y/dx2 • The derivative of a a second derivative: • The third derivative f’’’(x) = y’’’

  8. f(x) = x3 – 5x2 + 2 • f’(x) = • f’’(x) = • f’’’(x) = • f’’’’(x) =

  9. POWER RULE - REVISITED • f(x) = xn f’(x) = nxn-1 • What happens if I have a function raised to a power? • f(x) = (3x + 2)2 f’(x) = • f(x) = 9x2 + 12x + 4 f’(x) = 18x + 12 • We are going to change the rule to make it apply to other power situations • f(x) = un f’(x) = nun-1(du/dx) • f’’(x) = 2(3x + 2)1(3) = 6(3x + 2) = 18x + 12

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