What is the Axis of Symmetry?

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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

Solve 3(x - 2) < 18

vladimir2022 [97]

Hi there!

$\large\boxed{x < 8}$

3(x - 2) < 18

Begin by dividing both sides by 3:

3(x - 2)/3 < 18/3

x - 2 < 6

Add 2 to both sides:

x - 2 + 2 < 6 + 2

x < 8

5 0

2 years ago

Read 2 more answers

Consider the absolute value inequality: |x - 6| > 20. Will the graph of the solutions to the inequality result in an "and" or

Dmitrij [34]

Step-by-step explanation:

Case 1 : |x| > a, => x > a or x < -a

Case 2 : |x| < a, => -a < x < a

Since this question follows Case 1, we will have an "or" inequality.

3 0

3 years ago

Which expression is equivalent to 1/3 divided by 1/6

leonid [27]

Answer:

The answer is A

Step-by-step explanation:

cross mutliply

1×6=6

1×3=3

divide the answers

result: 2

I hope this helps

5 0

2 years ago

Jess deposited $5,000 into an account that earns simple interest. After 9 years, Jess had earned $3,150 in interest. What was th

musickatia [10]

Answer:

The interest rate of Jess's account was 7%

Step-by-step explanation:

A = P * (1 + rt)

A = final amount

P = initial principal balance

r = annual interest rate

t = time (in years)

Replacing with the values we know:

A = P * (1 + rt)

8,150 = 5,000 * (1 + r * 9)

8,150/5,000 = 1 + 9r

1.63 = 1 + 9r

1.63 - 1 = 9r

0.63 = 9r

r = 063/9

r = 0.07 = 7%

The interest rate of Jess's account was 7%

4 0

3 years ago

Read 2 more answers