Find the derivative of f(x)=√x by using the definition of derivative f ' (x)= lim h-->0 f(x+h)-f(x)/h?

Question

1) Find f(x+h)
I got √x+h

2) Find f(x+h)- f(x)
I got √x+h - √x

3) Find f(x+h)- f(x)/h
I got as far as: √x+h - √x /h

4) lim h-->0 f(x+h)- f(x)/h

Please when showing work can you keep it simple? For instance, for #3, tell me how I divide by h and what I need to cancel out if necessary along with the correct answer. I appreciate it and will award best answer!

Answer 1

f(x) = √x



1)

f(x + h) = √(x + h)



2)

f(x + h) - f(x) = √(x + h) - √(x)



3)

[f(x + h) - f(x)]/h

= [√(x + h) - √x]/h

= [√(x + h) - √x]/h * [√(x + h) + √x]/[√(x + h) + √x]

= [√(x + h)^2 - √x^2]/[h (√(x + h) + √x)]

= (x + h - x)/[h (√(x + h) + √x)]

= h/[h (√(x + h) + √x)]

= 1/[(√(x + h) + √x]



4)

lim(h->0) 1/[(√(x + h) + √x]

= 1/[(√(x + 0) + √x]

= 1/(2√x)



I hope this helps.

Answer 2

Difference of squares: a^2 - b^2 = (a+b)(a-b)

Using this in reverse is handy in this problem if you take √(x+h) - √x as (a-b).

Multiply numerator and denominator by (a+b) then:

(√(x+h) - √x)(√(x+h) + √x) / (h (√(x+h) + √x))

Following difference of squares, the numerator becomes a^2 - b^2. Since we made a √(x+h) and b √x, the numerator is x + h -x which is just h.

The denominator is just h(√(x+h) + √x)

Divide numerator and denominator by h to get:

1/ (√(x+h) + √x)

now, as the derivative is true as h approaches zero (your limit), make h zero.

1/(√x+√x)

so 1/(2√x)

is the derivative of √x.

Answer 3

　

You are completely ignoring the use of parentheses. This makes your answers incorrect:



f(x) = √x



1)



Find f(x+h) = √(x+h)



2)



Find f(x+h)- f(x) = √(x+h) - √x



3)



I very much doubt you were asked to find f(x+h)- f(x)/h

Most likely you were asked to find:

f(x+h) - f(x)

---------------

..... h

which is equivalent to (f(x+h) - f(x))/h = (√(x+h) - √x) / h



4)



lim [h→0] (f(x+h) − f(x)) / h

= lim [h→0] (√(x+h) − √x) / h

= lim [h→0] (√(x+h) − √x) (√(x+h) + √x) / (h (√(x+h) + √x))

= lim [h→0] ((√(x+h))² − (√x)²) / (h (√(x+h) + √x))

= lim [h→0] ((x+h) − x) / (h (√(x+h) + √x))

= lim [h→0] h / (h (√(x+h) + √x))

= lim [h→0] 1 / (√(x+h) + √x)

= 1 / (√x + √x)

= 1 / (2√x)

Answer 4

u will find the sol here https://socratic.org/questions/how-do-you-find-the-derivative-of-y-sqrt-x-using-the-definition-of-derivative

Answer 5

lim h-->0 [f(x+h)- f(x)]/h

[f(x+h)- f(x)]/h=[√x+h - √x] /h = [√x+h - √x] [√x+h + √x] /{h [√x+h + √x] } = 1/[√x+h + √x]



hence lim h-->0 1/[√x+h + √x] = ...