Q: Use the definition of a derivative to take the derivative of f(x) = 6sqrt(x)
Answer: First, let’s start with the definition of a derivative:
f ‘ (x) = lim (as h goes to 0) of [f(x+h) – f(x)] / h
So, using our given equation: f(x) = 6sqrt(x), we have:
f(x+h) = 6sqrt(x+h)
f(x) = 6sqrt(x)
So, the derivative is:
Step 1: The set-up) f ‘ (x) = limit as h goes to 0 of [6sqrt(x+h) – 6sqrt(x)]/h
Now, we need to do some simplifying before plugging in 0 for h.
The general strategy to use when you see square roots is to multiply both top and bottom of the fraction by the conjugate, which is (6sqrt(x+h) + 6sqrt(x):
Step 2: Simplifying) [6sqrt(x+h) – 6sqrt(x)] * (6sqrt(x+h) + 6sqrt(x) ) / h * (6sqrt(x+h) + 6sqrt(x) )
Simplify this more and nice things happen (distribute / FOIL the top):
36(x+h) – 36(x) / [h *(6sqrt(x+h) + 6sqrt(x))]
Simplify numerator more:
(36x + 36h – 36x) / [h *(6sqrt(x+h) + 6sqrt(x))]
36h / [h *(6sqrt(x+h) + 6sqrt(x))]
Cancel out “h”
36 / (6sqrt(x+h) + 6sqrt(x))]
Now, take the limit as h goes to 0 (plug in 0 for h):
36 / (6sqrt(x+0) + 6sqrt(x))] =
36 / (6sqrt(x) + 6sqrt(x))
36 / 12 sqrt(x)
Final answer: 3/sqrt(x)