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2 years ago

Please hurry I will give brainless answer if correct

Mathematics

1 answer:

Natali [406]2 years ago

7 0

Answer:

we have to be able to read it first tho. we can't see what the problem is so we need to see it please and thank you...

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Morgarella [4.7K]

Answer:

$f'(x)=-\frac{2}{x^\frac{3}{2}}$

Step-by-step explanation:

So we have the function:

$f(x)=\frac{4}{\sqrt x}$

And we want to find the derivative using the limit process.

The definition of a derivative as a limit is:

$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

Therefore, our derivative would be:

$\lim_{h \to 0}\frac{\frac{4}{\sqrt{x+h}}-\frac{4}{\sqrt x}}{h}$

First of all, let's factor out a 4 from the numerator and place it in front of our limit:

$=\lim_{h \to 0}\frac{4(\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt x})}{h}$

Place the 4 in front:

$=4\lim_{h \to 0}\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt x}}{h}$

Now, let's multiply everything by (√(x+h)(√(x))) to get rid of the fractions in the denominator. Therefore:

$=4\lim_{h \to 0}\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt x}}{h}(\frac{\sqrt{x+h}\sqrt x}{\sqrt{x+h}\sqrt x})$

Distribute:

$=4\lim_{h \to 0}\frac{({\sqrt{x+h}\sqrt x})\frac{1}{\sqrt{x+h}}-(\sqrt{x+h}\sqrt x)\frac{1}{\sqrt x}}{h({\sqrt{x+h}\sqrt x})}$

Simplify: For the first term on the left, the √(x+h) cancels. For the term on the right, the (√(x)) cancel. Thus:

$=4 \lim_{h\to 0}\frac{\sqrt x-(\sqrt{x+h})}{h(\sqrt{x+h}\sqrt{x}) }$

Now, multiply both sides by the conjugate of the numerator. In other words, multiply by (√x + √(x+h)). Thus:

$= 4\lim_{h\to 0}\frac{\sqrt x-(\sqrt{x+h})}{h(\sqrt{x+h}\sqrt{x}) }(\frac{\sqrt x +\sqrt{x+h})}{\sqrt x +\sqrt{x+h})}$

The numerator will use the difference of two squares. Thus:

$=4 \lim_{h \to 0} \frac{x-(x+h)}{h(\sqrt{x+h}\sqrt x)(\sqrt x+\sqrt{x+h})}$

Simplify the numerator:

$=4 \lim_{h \to 0} \frac{x-x-h}{h(\sqrt{x+h}\sqrt x)(\sqrt x+\sqrt{x+h})}\\=4 \lim_{h \to 0} \frac{-h}{h(\sqrt{x+h}\sqrt x)(\sqrt x+\sqrt{x+h})}$

Both the numerator and denominator have a h. Cancel them:

$=4 \lim_{h \to 0} \frac{-1}{(\sqrt{x+h}\sqrt x)(\sqrt x+\sqrt{x+h})}$

Now, substitute 0 for h. So:

$=4 ( \frac{-1}{(\sqrt{x+0}\sqrt x)(\sqrt x+\sqrt{x+0})})$

Simplify:

$=4( \frac{-1}{(\sqrt{x}\sqrt x)(\sqrt x+\sqrt{x})})$

(√x)(√x) is just x. (√x)+(√x) is just 2(√x). Therefore:

$=4( \frac{-1}{(x)(2\sqrt{x})})$

Multiply across:

$= \frac{-4}{(2x\sqrt{x})}$

Reduce. Change √x to x^(1/2). So:

$=-\frac{2}{x(x^{\frac{1}{2}})}$

Add the exponents:

$=-\frac{2}{x^\frac{3}{2}}$

And we're done!

$f(x)=\frac{4}{\sqrt x}\\f'(x)=-\frac{2}{x^\frac{3}{2}}$

5 0

3 years ago

I’m really stuck on this one if you can help i’d really appreciate it

Lynna [10]

you're correct. its to take out the greatest common factor.

8 0

3 years ago

Read 2 more answers

Luis rolled a number cube 60 times. He rolled the number 6 four times. Which is most likely the cause of the discrepancy between

slega [8]

He did not perform enough trials. The more trials you perform, the closer it gets to the predicted outcome.

4 0

3 years ago

Read 3 more answers

How do we find 4and1/4 hours after 11:20pm?

tigry1 [53]

there are 60 minutes in 1 hour, so 1/4 of an hour is 60(1/4), namely 15 minutes.

11:20pm + 4 hours, is 11+4:20, namely 15:20, of course the time system only uses up to 12, so that has to be 3:20, and then we add the 15 minutes.

11+4: 20 + 15.........3:35am.

5 0

3 years ago

Given a line segment with endpoints A(16, 8) and B(1, 3) what are the coordinates of the

blondinia [14]

Answer:

The line segment partitioned two-fifths from A to B is (10,6)

Step-by-step explanation:

First point from A to B is (16,8)

than find the difference between A to B i.e B - A

(1,3)-(16,8) = (-15,-5)

To measure the (2/5) difference we will multiply (-15,-5) with (2/5) which is equal to (-6,-2)

Now Add the difference to the first coordinate (point A) gives

Point of division = (16,8)+(-6,-2)

Point of division = (16-6, 8-2)

Point of division = (10,6)

4 0

3 years ago