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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 h→0 [f(x+h) – f(x)] / h

So, using our given equation:  f(x) = 6sqrt(x), we have:

f(x+h) = 6√(x+h)

f(x) = 6√(x)

So, the derivative is:

Step 1:  The set-up)  f ‘ (x) = limit h→0 of [6√(x+h) – 6√(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 (6√(x+h) + 6√(x):

Step 2:  Simplifying)  [6√(x+h) – 6√(x)] * (6√(x+h) + 6√(x)  ) / h * (6√(x+h) + 6√(x)  )

Simplify this more and nice things happen (distribute / FOIL the top):

36(x+h) – 36(x) / [h *(6√(x+h) + 6√(x))]

Simplify numerator more:

(36x + 36h – 36x) / [h *(6√(x+h) + 6√(x))]

36h / [h *(6√(x+h) + 6√(x))]

Cancel out “h”

36 / (6√(x+h) + 6√(x))]

Now, take the limit as h goes to 0 (plug in 0 for h):

36 / (6√(x+0) + 6√(x))] =

36 / (6√(x) + 6√(x))

36 / 12 √(x)

Final answer:  3/√(x)

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