Learning Outcomes

  1. Know the derivative of constant function and power functions, respectively.
  2. Apply the sum and difference rules to combine derivatives.
  3. Use the product rule for finding the derivative of a product of functions.
  4. Use the quotient rule for finding the derivative of a quotient of functions.
  5. Combine the differentiation rules to find the derivative of a polynomial or rational function.

Textbook

  • Chapter 3.3   Differentiation Rules  

Textbook Assignment

  • p. 263:    107, 110, 112, 115-117 all

WeBWorK Assginment

  • Derivatives-Power Rule
  • Derivatives-Product Rule
  • Derivatives-Quotient Rule

Exit problems of the session 

  1. Find the derivative of the following functions:

(a).   f(x)=(x^2-1)(2x^3+x+2)   (b).   g(x)=\dfrac{x^2+6}{x^2-6}     (c).   h(x)=5x^4-\dfrac{3}{x^2}

  2.  Let  f(3)=4  and  f'(3)=-1.   Find  h'(3)   if   h(x)=2x^2\cdot f(x).

 

 Key Concepts

  • The derivative of a constant function is zero:  \dfrac{d}{dx}(c)=0.
  • Power Rule:    \dfrac{d}{dx}(x^n)=n\cdot x^{n-1}.
  • Constant Multiple Rule: if f(x)=c \cdot g(x), then if f'(x)=c \cdot g'(x).
  • Sum Rule: if h(x)=f(x)+g(x), then h'(x)=f'(x)+g'(x).
  • Difference Rule: if h(x)=f(x)-g(x), then h'(x)=f'(x)-g'(x).
  • Product Rule: if h(x)=f(x)g(x), then h'(x)=f'(x)g(x)+f(x)g'(x).
  • Quotient Rule: if h(x)=\dfrac{f(x)}{g(x)}, then h'(x)=\dfrac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}.
 

Videos and Practice Problems of Selected Topics

  • Basic Rules
  1. \rhd Basic derivative rules (Part 1)  (2:26) The derivative of a constant function, f(x) =k, using the limit definition and a geometric approach.
  2. * Practice: A summary of the differentiation rules – sum, difference, product and quotient. (4 problems)
  3. \rhd Basic derivative rules: table(8:59) A table with the values of f(x) and f'(x) are given for x=0, 1, 4, 9, 16.  The functions g and h are defined as g(x) = |x-1|+1 and h(x) = 3f(x)+2g(x). Find \dfrac{d}{dx}h(x) at x=9, or equivalently, h'(9).
  4. * Practice: Basic derivative rules: table. (4 problems)
  5. \rhd Power rule (3:53) The derivative of x^n when n\neq 0. Examples shown: x^2, x^3, x^{-100}, x^{2.572}.
  6. \rhd Basic derivative rules (9:52) The derivative of a sum of power functions or power functions multiplied by a constant.  Find the derivative of 2x^5, x^3+x^{-4} and 2x^3-7x^2+3x-100.
  7. \rhd Differentiating integer powers (mixed positive and negative) (4:48) Consider g(x) = \dfrac{2}{x^3} - \dfrac{1}{x^2}. Find g'(x) and g'(2).
  8. * Practice: Power rule (negative and fractional powers). (4 problems)
  9. \rhd Tangent to the graph of \dfrac{1}{x} at the point on the curve x=k (5:58)
  10. \rhd Fractional powers differentiation (2:35) Find h'(16) when h(x) = 5x^{1/4}+7.
  11. * Practice: Power rule (with rewriting the expression). (4 problems)
  12. \rhd Radical functions differentiation (2:39) Find f'(8) when f(x) = -4\sqrt[3]x.
  13. * Practice: Differentiate integer powers (mixed positive and negative). (4 problems)
  • Polynomials
  1. \rhd Differentiating polynomials (6:38) Find the derivative of f(x) = x^5+2x^3-x^2 and f'(2).
  2. * Practice: Differentiate polynomials. (4 problems)
  3. \rhd Tangents of polynomials (4:41) Consider f(x) = x^3 -6x^2+x-5. Find the equation of the tangent line to f(x) at x=1.
  4. * Practice: Tangents of polynomials. (4 problems)
  • Product Rule
  1. \rhd Product rule (8:03) Find the derivative of h(x) = (x^2)(x^3+4) (first 3:20 minutes) and y= (\sin x)(\cos x) (x^2+1) (can be skipped if you have not seen the derivative of trig functions).
  • Quotient Rule
  1. \rhd Quotient rule (7:37) Find the derivative of y=\dfrac{x^2+1}{x^5+x} (first 3:34 minues) and y=\dfrac{\tan x}{x^{3/2}+5x} (can be skipped if you have not seen the derivative of trig functions).