Learning Outcomes

- Know the derivative of constant function and power functions, respectively.
- Apply the sum and difference rules to combine derivatives.
- Use the product rule for finding the derivative of a product of functions.
- Use the quotient rule for finding the derivative of a quotient of functions.
- 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 **

- 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**

- $\rhd$ Basic derivative rules (Part 1) (2:26) The derivative of a constant function, $f(x) =k$, using the limit definition and a geometric approach.
- * Practice: A summary of the differentiation rules – sum, difference, product and quotient. (4 problems)
- $\rhd$ 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, $h'(9)$.
- * Practice: Basic derivative rules: table. (4 problems)
- $\rhd$ Power rule (3:53) The derivative of $x^n$ when $n\neq 0$. Examples shown: $x^2$, $x^3$, $x^{-100}$, $x^{2.572}$.
- $\rhd$ 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 $2x^3-7x^2+3x-100$.
- $\rhd$ Differentiating integer powers (mixed positive and negative) (4:48) Consider $g(x) = \dfrac{2}{x^3} – \dfrac{1}{x^2}$. Find $g'(x)$ and $g'(2)$.
- * Practice: Power rule (negative and fractional powers). (4 problems)
- $\rhd$ Tangent to the graph of $\dfrac{1}{x}$ at the point on the curve $x=k$ (5:58)
- $\rhd$ Fractional powers differentiation (2:35) Find $h'(16)$ when $h(x) = 5x^{1/4}+7$.
- * Practice: Power rule (with rewriting the expression). (4 problems)
- $\rhd$ Radical functions differentiation (2:39) Find $f'(8)$ when $f(x) = -4\sqrt[3]x$.
- * Practice: Differentiate integer powers (mixed positive and negative). (4 problems)

**Polynomials**

- $\rhd$ Differentiating polynomials (6:38) Find the derivative of $f(x) = x^5+2x^3-x^2$ and $f'(2)$.
- * Practice: Differentiate polynomials. (4 problems)
- $\rhd$ 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 $x=1$.
- * Practice: Tangents of polynomials. (4 problems)

**Product Rule**

- $\rhd$ 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**

- $\rhd$ 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).