2.2 Formal Derivative Problem Set
Use the formal definition of the derivative:
lim
∆𝑥→0
𝑓(𝑥+∆𝑥)−𝑓(𝑥)
𝑥+∆𝑥 − 𝑥
or lim
ℎ→0
𝑓(𝑥+ℎ)−𝑓(𝑥)
𝑥+ℎ − 𝑥
Or the alternate version. No magic yet.
𝑓(𝑥) − 𝑓(𝑐)
𝑥→𝑐
𝑥−𝑐
lim
Find the derivative of each of the following functions with respect to 𝑥.
𝑓(𝑥) = 4 − 2𝑥
𝑦 = 5𝑥 2 − 4𝑥
𝑔(𝑥) = √𝑥 + 4
𝑦 = √𝑥 2 − 3
Write the equation of the tangent line to the given function at the indicated x-value.
1
𝑓(𝑥) = 𝑥 2
at 𝑥 = 2
𝑦 = √𝑥
at 𝑥 = 4
ℎ(𝑥) = 3𝑥 2 − 5𝑥
at 𝑥 = 2
4
𝑦 =𝑥+𝑥
at 𝑥 = 4
𝑓(𝑥) = √𝑥 − 1
at 𝑥 = 5
𝑦 = 𝑥2 − 4
at 𝑥 = −2
Find an equation of the line that is tangent to the graph of the first function and is parallel to the given line.
𝑓(𝑥) = 2𝑥 2 − 1
𝑦=
1
2𝑥 − 5𝑦 = 4
−𝑥 = 2𝑦 − 5
√𝑥
Find an equation of the line that is tangent to the graph of the first function and is perpendicular to the given
line.
2
𝑘(𝑥) = 3 + 3 𝑥 2
−4𝑥 + 2𝑦 = 7
𝑦 = √𝑥 + 2
4𝑥 + 2𝑦 − 6 = 0