JAN 30

Derivatives and Functions

Introduction to Functions

• First Function: Defined as ( f(x) = 4x + 3 )

• Second Function: Defined as ( s(x) = x^2 - 2x )

Derivatives of Functions

• Finding the Derivative:

  • The derivative of the first function, ( f' )

  • Result: ( f' = 4 )

  • The derivative of the second function, ( s' )

  • Result: ( s' = 2x - 2 )

Combining Derivatives

• Product Rule: To find the derivative of the product of two functions, use:
  [ f' = f imes s' + s imes f' ]

• Substitute the functions and their derivatives:

  • ( f' = (4x + 3)(2x - 2) + (x^2 - 2x)(4) )

Simplifying the Resulting Expression

• Distributing Terms: Break down each part:

  • ( 4x \times 2x = 8x^2 )

  • ( 4x \times (-2) = -8x )

  • ( 3 \times 2x = 6x )

  • ( 3 \times (-2) = -6 )

  • For the second term:

  • ( x^2 \times 4 = 4x^2 )

  • ( -2x \times 4 = -8x )

• Combining Like Terms:

  • Combine ( 8x^2 ) and ( 4x^2 )

  • Result: ( 12x^2 )

  • Combine ( -8x + 6x - 8x )

  • Result: ( -10x )

  • Combine constants:

  • Result: ( -6 )

• Final Answer: The derivative simplified is:
  [ 12x^2 - 10x - 6 ]

Finding the Slope of a Tangent Line

• Given Function: ( y = x^3 - 1 )

• Step to Find Slope:

  • The slope at a point is determined by finding the derivative at a specific ( x ) value.

• Second Function: ( g(x) = x^2 + 4x - 5 )

  • Individual derivatives:

  • ( (x^3 - 1)' = 3x^2 )

  • ( (x^2 + 4x - 5)' = 2x + 4 )

Using the Product Rule in Slope Calculation

• Combining Derivatives:
  [ y' = (x^3 - 1)' imes (g(x)) + (x^3 - 1)(g'(x)) ]

• Simplification Points:

  • If evaluating at specific ( x ) value, plug in before simplifying.

• Example evaluation at ( x = 2 ):

  • Substitute each term and compute without full expansion.

Key Points on Algebra in Calculus

• Algebra Skills: Important to maintain proficiency in algebra for simplifying derivatives, especially when combining terms.

• Product and Quotient Rules:

  • For products: ( u imes v' + v imes u' )

  • For quotients (Quotient Rule): [ \frac{v \times u' - u \times v'}{v^2} ]

Revisiting Quotient Rule Derivation

• Example Derivation:

  • Top: ( u = \sqrt{x} + 3 )

  • Bottom: ( v = 4x - 1 )

  • Rewrite ( u = (x)^{1/2} + 3 )

• Calculating Derivative:

  • Top derivative: ( u' = \frac{1}{2}x^{-1/2} )

• Final Expression for Derivative of a Quotient:
  [ \frac{(4x - 1)(u') - (\sqrt{x} + 3)(4)}{(4x - 1)^2} ]

• Plugging In Values: Whenever possible, plug in values directly to avoid complex symbolic algebra wherever appropriate.

Common Mistakes to Avoid

• Be cautious of the negative signs in derivatives as these can lead to calculation errors.

• Always distribute negatives appropriately across terms in the quotient rule.

Assessment and Application

• Students are encouraged to practice finding derivatives through multiple examples and identify common patterns to ease calculations across subjects.