1. 3.1 –Tangents and the Derivative at a Point
Defn: Derivative:
The slope of the tangent line to the graph of a function at a given point. 
The instantaneous rate of change of a function with respect to its variable.
The limiting value of the ratio of the change in a function to the corresponding change in its independent variable.
𝑓 ′ 𝑥 𝑟𝑒𝑎𝑑𝑠,"𝑓 𝑝𝑟𝑖𝑚𝑒 𝑜𝑓 𝑥."

2. 3.1 –Tangents and the Derivative at a Point

3. 3.1 –Tangents and the Derivative at a Point
Instantaneous Rate of Change / the Slope of a Tangent Line at a Point
𝑓 𝑥 =−2 𝑥 ,2
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim ℎ→0 ℎ −4−2ℎ ℎ
𝑚 𝑡𝑎𝑛 =𝑓′(𝑥)= lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 𝑥+ℎ −𝑥 =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim ℎ→0 −4−2ℎ
𝑚 𝑡𝑎𝑛 =𝑓′(1)= lim ℎ→0 𝑓 1+ℎ −𝑓 ℎ −1 =
𝑚 𝑡𝑎𝑛 =𝑓′(1)=−4
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim ℎ→0 −2 1+ℎ 2 +4− − ℎ =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim ℎ→0 −2 1+2ℎ+ ℎ 2 +4− − ℎ
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim ℎ→0 −2−4ℎ−2 ℎ 2 +4−2 ℎ
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim ℎ→0 −4ℎ−2 ℎ 2 ℎ

4. 3.1 –Tangents and the Derivative at a Point
Alternate Definition for the Derivative at a Point

5. 3.1 –Tangents and the Derivative at a Point
Instantaneous Rate of Change / the Slope of a Tangent Line at a Point
𝑓 𝑥 =−2 𝑥 ,2
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim 𝑥→1 −2 𝑥+1 𝑥−1 𝑥−1
𝑚 𝑡𝑎𝑛 =𝑓′(𝑎)= lim 𝑥→𝑎 𝑓 𝑥 −𝑓 𝑎 𝑥−𝑎
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim 𝑥→1 −2 𝑥+1
𝑚 𝑡𝑎𝑛 =𝑓′(1)= lim 𝑥→1 𝑓 𝑥 −𝑓 1 𝑥−1 =
𝑚 𝑡𝑎𝑛 =𝑓′(1)=−4
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim 𝑥→1 −2 𝑥 2 +4− − 𝑥−1 =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim 𝑥→1 −2 𝑥 2 +4−2 𝑥−1
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim 𝑥→1 −2 𝑥 2 +2 𝑥−1
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim 𝑥→1 −2 𝑥 2 −1 𝑥−1

6. 3.2 –The Derivative of a Function
Differentiation is the process used to develop the derivative.
Differentiating a function will create the derivative.

7. 3.2 –The Derivative of a Function
𝐸𝑥𝑎𝑚𝑝𝑙𝑒: 𝑓 𝑥 = 𝑥 2 +𝑥+1
𝑚 𝑡𝑎𝑛 =𝑓′(𝑥)= lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim ℎ→0 𝑥+ℎ 2 + 𝑥+ℎ +1− 𝑥 2 +𝑥+1 ℎ =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim ℎ→0 𝑥 2 +2𝑥ℎ+ ℎ 2 +𝑥+ℎ+1− 𝑥 2 −𝑥−1 ℎ =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim ℎ→0 2𝑥ℎ+ ℎ 2 +ℎ ℎ =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim ℎ→0 ℎ 2𝑥+ℎ+1 ℎ =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim ℎ→0 2𝑥+ℎ+1
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 =2𝑥+1

8. 3.2 –The Derivative of a Function
𝐸𝑥𝑎𝑚𝑝𝑙𝑒: 𝑓 𝑥 = 𝑥 2 +𝑥+1
𝑚 𝑡𝑎𝑛 =𝑓′(𝑥)= lim 𝑎→𝑥 𝑓 𝑎 −𝑓 𝑥 𝑎−𝑥
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎+𝑥+1
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎 2 +𝑎+1− 𝑥 2 +𝑥+1 𝑎−𝑥 =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 =𝑥+𝑥+1
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎 2 +𝑎+1− 𝑥 2 −𝑥−1 𝑎−𝑥 =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 =2𝑥+1
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎 2 +𝑎− 𝑥 2 −𝑥 𝑎−𝑥 =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎 2 − 𝑥 2 +𝑎−𝑥 𝑎−𝑥 =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎+𝑥 𝑎−𝑥 + 𝑎−𝑥 𝑎−𝑥 =
𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎−𝑥 𝑎+𝑥+1 𝑎−𝑥