Find the requested derivative using limits

a. 𝑓′(-4)if f(x)=sqrt(1-2x)

b. 𝑓′(a)if f(x)=(2x+1)/(x+3)

c. 𝑓′(x) if f(x)=x^-2

These are a bit tedious, but not hard. I'm sure your book has examples just like them. Give it a go, and come back if you get stuck. Show what you did.

Just recall that √(1-2(x+h)) - √(1-2x)
can be "rationalized" by multiplying top and bottom by √(1-2(x+h)) + √(1-2x).

That way the 0/0 is avoided, and you then have a nonzero factor in the denominator.

Steve I agree they are tedious doing it this way its easier using the derivative rules but even though i know them we arent allowed to use them until the class learns them. :(

To find the derivative using limits, we can use the definition of the derivative:

𝑓′(𝑥) = lim(𝑥→𝑥₀) [𝑓(𝑥)−𝑓(𝑥₀)] / (𝑥−𝑥₀)

a. 𝑓′(−4) if 𝑓(𝑥) = √(1−2𝑥):

Step 1: Start with the given function 𝑓(𝑥) = √(1−2𝑥).
Step 2: Determine 𝑓(𝑥₀) by substituting −4 into the function:
𝑓(−4) = √(1−2(−4)) = √9 = 3.
Step 3: Substitute the values into the derivative formula:
𝑓′(−4) = lim(𝑥→−4) [𝑓(𝑥)−𝑓(−4)] / (𝑥−(−4)).
Step 4: Simplify the expression:
𝑓′(−4) = lim(𝑥→−4) [√(1−2𝑥)−3] / (𝑥+4).

b. 𝑓′(𝑎) if 𝑓(𝑥) = (2𝑥+1) / (𝑥+3):

Step 1: Start with the given function 𝑓(𝑥) = (2𝑥+1) / (𝑥+3).
Step 2: Determine 𝑓(𝑎) by substituting 𝑎 into the function:
𝑓(𝑎) = (2𝑎+1) / (𝑎+3).
Step 3: Substitute the values into the derivative formula:
𝑓′(𝑎) = lim(𝑥→𝑎) [𝑓(𝑥)−𝑓(𝑎)] / (𝑥−𝑎).
Step 4: Simplify the expression:
𝑓′(𝑎) = lim(𝑥→𝑎) [(2𝑥+1) / (𝑥+3) − (2𝑎+1) / (𝑎+3)] / (𝑥−𝑎).

c. 𝑓′(𝑥) if 𝑓(𝑥) = 𝑥^(−2):

Step 1: Start with the given function 𝑓(𝑥) = 𝑥^(−2).
Step 2: Determine 𝑓(𝑥₀) by substituting 𝑥 into the function:
𝑓(𝑥₀) = 𝑥₀^(−2).
Step 3: Substitute the values into the derivative formula:
𝑓′(𝑥) = lim(𝑥→𝑥₀) [𝑓(𝑥)−𝑓(𝑥₀)] / (𝑥−𝑥₀).
Step 4: Simplify the expression:
𝑓′(𝑥) = lim(𝑥→𝑥₀) [𝑥^(−2) − 𝑥₀^(−2)] / (𝑥−𝑥₀).

Note: In each case, you may need to perform additional simplifications or algebraic manipulations if required.