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Review of Derivatives

Review of Derivatives. Power rule, product rule, quotient rule and chain rule. The Power Rule. Remember, the power rule only works on functions of the form y = x n . The power rule says that y’ = nx n-1 Examples: y = x 2 , so y’ = 2x y =x 1/2 , so y’ = ½x -1/2 y = x -1 , so y’ = -x -2.

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Review of Derivatives

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  1. Review of Derivatives Power rule, product rule, quotient rule and chain rule

  2. The Power Rule • Remember, the power rule only works on functions of the form y = xn. • The power rule says that y’ = nxn-1 • Examples: • y = x2, so y’ = 2x • y =x1/2, so y’ = ½x-1/2 • y = x -1, so y’ = -x -2

  3. The Product Rule • The product rule can be used when two functions are multiplied together. • If y = f(x)g(x), then y’ = f’(x)g(x) + f(x)g’(x) • Examples: • If y = xsinx, then y’ = sinx +xcosx • If y = (3x)(5x+1), then y’ = (3)(5x+1) + (3x)(5) • Of course, you must remember to simplify your answers!

  4. The Quotient Rule • The quotient rule can be used when two functions are being divided. • If y = f(x)/g(x), then y’ = [g(x)f’(x) – f(x)g’(x)]/(g(x))2, or (lodhi – hidlo) / lolo !  • Example: • If y = sinx/cosx, then y’ = [cosx(cosx) – sinx(-sinx)]/cos2x • What does this simplify to???

  5. Trigonometric Derivatives • If y = sinx, then y’ = cosx • If y = cosx, then y’ = -sinx • If y = tanx, then y’ = sec2x • If y = secx, then y’ = secxtanx • If y = cscx, then y’ = -cscxcotx • If y = cotx, then y’= -csc2x

  6. The Chain Rule • The chain rule is used on composition functions. • You must identify the inside function and the outside function. • The chain rule says if y = f(g(x), then y’ = f’(g(x))*g’(x), or the derivative of the inside times the derivative of the outside

  7. The Chain Rule (cont’d) • Examples: • If y = sin(x2), then y’ = 2xcos(x2) • If y = (2x+1)3 then y’ = 2*3(2x+1)2 • Remember, the product rule and the quotient rule may also need to be used along with the chain rule!! • If y = (2x+1)3(3x+2)2, then y’ = 2*3(2x+1)2(3x+2)2 + (2x+1)3(3)(2(3x+2)) • Don’t forget to simplify!!!

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