# How do you find the derivative of y =sqrt(x) using the definition of derivative?

y= ##sqrt x##
y=##x^(1/2)##

Don't use plagiarized sources. Get Your Custom Essay on
How do you find the derivative of y =sqrt(x) using the definition of derivative?
Just from \$13/Page

Differentiate w.r.t “x” on both sides:

##dy/dx= d/dx [x^(1/2)]##

##dy/dx= 1/2x^(1/2-1)## (because ##d/dx[x^n]=nx^(n-1)##)

##dy/dx= 1/2x^(-1/2)##

And it can also be written as:

##dy/dx= 1/(2sqrt(x))##

Or, if you meant the limit definition of the derivative function it would look like this:

##f'(x) = lim_(h->0)(f(x+h)-f(x))/h##

##f'(x) = lim_(h->0)(sqrt(x+h)-sqrtx)/h##

Now, we multiply the numerator and the denominator by the conjugate of the numerator (conjugates are the sum and difference of the same two terms such as a + b and a – b).

##f'(x) = lim_(h->0)(sqrt(x+h)-sqrtx)/h*(sqrt(x+h)+sqrtx)/(sqrt(x+h)+sqrtx)##

Since ## (a+b)(a-b) = a^2-b^2## we get

##f'(x)=lim_(h->0)(x+h-x)/(h(sqrt(x+h)+sqrtx)##

Simplifying, we get

##f'(x)=lim_(h->0)h/(h(sqrt(x+h)+sqrtx)##

##f'(x)=lim_(h->0)1/(sqrt(x+h)+sqrtx)##

If we evaluate the limit by plugging in ##0## for ##h## we get

##f'(x)=1/(sqrt(x+0)+sqrtx)=1/(sqrtx+sqrtx)=1/(2sqrtx)##

## Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
\$26
The price is based on these factors:
Number of pages
Urgency
Basic features
• Free title page and bibliography
• Unlimited revisions
• Plagiarism-free guarantee
• Money-back guarantee
On-demand options
• Writer’s samples
• Part-by-part delivery
• Overnight delivery
• Copies of used sources
Paper format
• 275 words per page
• 12 pt Arial/Times New Roman
• Double line spacing
• Any citation style (APA, MLA, Chicago/Turabian, Harvard)

# Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

### Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

### Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

### Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.