Question

Use the definition of the derivatives to differentiate f(x) = 5√x

Answer 1

Hello Gawain!
Welcome to OpenStudy! :)

Answer 2

hello

Answer 3

For the limit definition of our derivative we'll have two pieces in the numerator,\[\large\rm f(\color{orangered}{x})=5\sqrt{\color{orangered}{x}}\]\[\large\rm f(\color{orangered}{x+h})=5\sqrt{\color{orangered}{x+h}}\]You ok with that second piece?
It's just the function evaluated at x+h instead of x.

Answer 4

yes

Answer 5

So the limit definition of the derivative is just the slope formula, or difference quotient, and we're letting the space between the two points get smaller and smaller.
That's what our limit is doing.\[\large\rm \lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]We'll plug in our pieces,\[\large\rm \lim_{h\to0}\frac{5\sqrt{x+h}-5\sqrt{x}}{h}\]

Answer 6

When dealing with limits, try to approach them with this thought process:
1. Plug the value directly into the function.
2. If step 1 causes problem, do some algebra, try to `cancel` something.
3. Repeat step 1.

Answer 7

If we plug h=0 directly in, it causes a problem ya?
It puts a 0 in our denominator.
We can't divide by 0 in the land of math.

So we back up, and try to do some algebra steps.

One approach would be to use `conjugates`.
Recall this: \(\large\rm (a-b)(a+b)=a^2-b^2\)

Any idea how we can maybe make use of that? :d
Hmm what do you think? :)

Answer 8

thank you, it is very helpful, and i understood it :)