Using the Definition of a Derivative…

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)

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