Definition of the Derivative.?

Question

Use the definition of the derivative to find the derivative of:
a) -(2/3x)

b) (sq. rt.2x) - x^3

how many times do i have tell people i check my answers against others. its NOT, again NOT my homework.
due to the negative powers i used i want to make sure my working is exact.

well... its -2/(3x)
i did:
f'(x) = lim -2(3(x+h))^-1 + 2(3x)^-1 / h
and then i cant go on or or do

f'(x) = lim (-2/(3x)) - (2/(3h)) + (2/(3x)) / h

ooooo please help me :(

Answer 1

derivative just means instantaneous rate of change

a) I'm going to assume you meant -( (2/3)x). In this case, the derivative is -2/3.

if it is -(2 / (3x) then use quotient rule:

y' (u/v) = (vu' - uv' )/v^2

y = -2/(3x)

y' = [ (3x) d/dx(-2) - (-2) d/dx(3x) ] (3x)^2

y' = ((3x * 0) + 2*3 ) / 9x^2

y' = 6/(9x^2)

y' = 2/(3x^2)


b) y = sqrt(2x) - x^3

y' = d/dx (sqrt(2x)) + d/dx (-x^3)

y' = d/dx (2x)^(1/2) + d/dx (-x^3)

y' = 1/2 * (2x)^(-1/2) d/dx(2x) - 3x^2

y' = 2/(2 (2x)^(1/2)) - 3x^2

y' = 1 / (sqrt(2x)) - 3x^2

Answer 2

the definition of a derivative is a little different than actually taking the derivative..........the definition is:

y' = lim [ f ( x + h ) - f ( x ) ] / h , ........ as the limit h ==> 0
[ h ==>0 , means as h approaches zero ]

for example: f ( x ) = - 2 x / 3 [ assuming you meant ( -2 / 3 ) x ].........then f ( x + h ) = ( - 2/3 ) ( x + h ) and the [ ] above gives........... [ ( -2/3 ) (x + h ) - ( -2/3)x ] = ( -2/3)h.
Then dividing by h, you have -2/3 ...... taking the limit of -2/3 as h ==>0 , you will get -2/3....(( notice he h vanished from this simple problem ))

The [ f ( x + h ) - f ( x ) ] / h is called the " difference quotient", and is an essential part of the Definition of the derivative.

Applying the definition to harder problems, such as sq roots and powers, takes more work, so I hope you don't really have to do it this way!!!!!

Answer 3

My understanding of problem is that "Use the definition of the derivative to find the derivative of" means use:

f'(x) = dy/dx = lim h ->0 (f(x +h) - f(x))/h

Show your work and I can comment

Answer 4

(-2/(3(x + h) + 2/(3x))/h =
(2/3)(1/x - 1/(x + h))/h =
(2/3)((x + h - x)/(x(x + h))/h =
(2/3)(h/(xh(x + h)) =
(2/3)(1/(x^2 + hx)) =
lim(2/3)(1/(x^2 + hx)) =
h→0
2/(3x^2)

[(2(x + h))^(1/2) - (x + h)^3 - (2x)^(1/2) + x^3]/h =

[(2(x + h) - 2x) / (h(2(x + h))^(1/2) + (2x)^(1/2))) + (x^3 - x^3 - 3x^2h - 3xh^2 - h^3)/h] =

[(2x + 2h - 2x) / (h(2(x + h))^(1/2) + (2x)^(1/2))) + (- 3x^2h - 3xh^2 - h^3)/h] =

[2 / (2(x + h))^(1/2) + (2x)^(1/2)) - 3x^2 - 3xh - h^2)]

lim[2 / (2(x + h))^(1/2) + (2x)^(1/2)) - 3x^2
h→0
- 3xh - h^2)] =

[2 / (2x)^(1/2) + (2x)^(1/2)) - 3x^2] =
[2 / (2(2x)^(1/2)) - 3x^2] =

1 / (2x)^(1/2) - 3x^2

Answer 5

do ur own homework.. this stuff is so simple.. if u can't do this, u're pretty much screwed..