Help me with this Derivative Question (I have the answer but don't know how to solve it)?

Question

The Question is:
What is the value of lim((x + "delta" x)^3 - x^3)/ "delta" x
if x=3
The answer is 3x^2 and thus 27.
How do you arrive at this answer? Can someone explain to me how to solve these kinds of questions?

Answer 1

Ok I am assuming you mean the limit as "delta" x approaches 0.

This is the basic formula for the derivative

Lim(((f(x+"delta" x) - f(x))/"delta" x)

To solve this particular problem focus on the numerator first

You need to expand the (x + "delta" x)^3

thus the numerator looks like the following

(x+ "delta" x)^3 - x^3
= (x^2 + 2x*"delta x" + "delta x"^2)*(x + "delta x") - x^3
=x^3 + 3x^2*"delta x" + 3x*"delta x"^2 + "delta x"^3 - x^3

By simplifying your numerator becomes

3x^2*"delta x" + 3x*"delta x"^2 + "delta x"^3

All of this is divided by "delta x" and since there is a "delta x" in each section of the numerator you can cancel one out leaving your limit to be the following

lim 3x^2 + 3x*"delta x" + "delta x"^2

Since this is the limit as "delta x" approaches 0 you can plug in 0 for all "delta x" 's and so you get

lim 3x^2 + 3x*"delta x" + "delta x"^2
= 3x^2

Since x=3 then obviously the answer is 27.

Hope this helps!!

Answer 2

This is an application of the limit definition for a derivitive, the function involved is y = x^3.
The approach is to expand the (x+ del x)^3 term, and then subtract x^3, which leaves a whole bunch of x^2delx, x*delx^2 and delx^3 terms in the numerator. As del x becomes smaller and smaller, all the terms except the 3x^2 del x term vanish at the limit. Since this is divided by del x, the answer (derivitive of x^3), is 3x^2.

Answer 3

This the definition of the derivative of x^3 at x = 3.

the derivative is 3x^2, and the value of this at x = 3 is 27.

Answer 4

Indeed, this is the definition of the derivative, but you can just do a straight-out calculation!

Expand: let's call delta x = h

[(x+h)^3 - x^3] / h or

[ (x^3 + 3h^2x + 3hx^2 + h^3) - x^3]/h, simplify

[ 3h^2x + 3hx^2 + h^3 ]/h, cancel h

3hx + 3x^2 + h^2 , now let h (delta x) go to zero, and you have

3x^2, the derivative.

Answer 5

delta x will be denoted by h
( (x + h)^3 - x^3 )/h
( x^3 + 3x^2h + 3xh^2 + h^3 - x^3 )/h
( 3x^2h + 3xh^2 + h^3 )/h
3x^2 + 3xh + h^2
lim[h -> 0] 3x^2 + 3x(0) + 0^2
3x^2

Answer 6

First expand (x+Δx)^3

(x+Δx)^3 = x^3 +(Δx)^3 + 3x^2 Δx + 3x (Δx)^2

subtract x^3

(x+Δx)^3 - x^3 = x^3 +(Δx)^3 + 3x^2 Δx + 3x (Δx)^2 -x^3

=>(Δx)^3 + 3x^2 Δx + 3x (Δx)^2

take out Δx common

=>Δx[(Δx)^2 + 3x^2 + 3x Δx]

now divide by Δx

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

=>(Δx)^2 + 3x^2 + 3x Δx

so when (Δx) approaches zero, the limit of the the given expression becomes

0 + 3x^2 + 3x(0)

=> 3x^2

Answer 7

[x+dx][x+dx]=x^2+2xdx+dx^2 => * x+dx=
x^3+x^2dx+2x^2dx+2xdx^2+xdx^2+dx^3 subtract x^3 and then divide by dx
x^2 +2x^2+2dx+xdx+dx^2 Now set dx = 0 and all that is left is x^2+2x^2=3x^2