easy derivative question?

Question

please explain to me using the power rule, how this answer is given:

d/dx (1/x) = d/dx (x ^ -1)

i don't understand how the x got in the front. is the x considered the n in the power rule equation:

d/dx (x ^ n) = nx ^ n-1

it goes on to say:

-1x ^ -2 = -1/x ^ 2

i don't understand the -1 divided by x^2. are they simply dividing the eequation by x^2 to get the -1 on top?

Answer 1

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

Answer 2

You need to go and revise all your basic maths about exponentiation. Maybe revising fractions will help too.

The rule for derivatives of powers is:
d/dx (x^n) = n (x^(n-1))

So if n = -1, just substitute and you get
d/dx (x^-1) = -1 (x^-2)

Now you have to understand what negative powers are.
The rule for multiplying powers of x are:
x^n x^m = x^(n+m)

For example,
x^2 x^3 = x*x * x*x*x = x^5 (*=multiply)

Now suppose you have
x^n x^m = 1
Given some value of n, what value of m would make x^n x^m = 1?
Well, 1 = x^0, so
x^n x^m = x^(n+m) = x^0.
So n+m = 0
m = -n.
Negative power x^-n means something you multiply with x^n to get 1.
But there's something else you multiply by x^n to get 1, and that's 1/(x^n):
x^n 1/x^n = x^n / x^n = 1.
So in other words, x^-n is the same thing as 1/(x^n)

So d/dx (1/x) = d/dx (x^-1) because x^-1 = 1/(x^1) = 1/x.

Similarly, the derivative has x^-2. This is the same as 1/(x^2).
So - x^-2 = (-1) (x^-2) = (-1) (1/x^2) = -1/(x^2).

Answer 3

You need to understand how powers work.
The power will change sign when it is brought above or below the division line. Plus will change to minus and minus will change to plus.

x², if this goes below the line it becomes x^-².
1/ x^-3, if this goes above the line it becomes x^3.

Now let's look at what you have.
d/dx (1/x) = d/dx (x ^ -1)
Divide both sides by d/dx and you get:
(1)(1/x) = (1)(x ^ -1) But one of anything is the anything.
(1/x) = (x ^ -1)
Now this matches up with what I said above.


Example.
f(x) = 1/x²
d/dx = 1/x² dx
d/dx = x^-2 dx
Now, multiply by the power and then reduce the power by one.
d/dx = x^-2 dx
d/dx = (-2)x^-3
To make the minus power a plus, bring it below the line.
d/dx = (-2)/ x^-3
d/dx = -2/ x^3

You started off with:
d/dx = 1/x² dx
And you ended up with:
d/dx = -2/ x^3
And this matches the formula:
d/dx (x ^ n) = nx ^ n-1
d/dx (x ^ n) = n/x ^ n+1

Answer 4

d/dx (1/x)
d/dx (x ^ -1) ==> here x goes up to the numerator part, therefore its power's sign will be reverse. ie (1 to -1)

By power rule:
d/dx (x ^ n) = nx ^ n-1
d/dx (x ^ -1) = -1 * x ^(-1-1)

-1 * x ^ -2 = -1 / x ^ 2 ==> here x^-2 is brought to the denominator part, therefore the -2 exponent will become +2.

ans: [ -1 / x^2 ]

Answer 5

1/x=x^-1
that is any number on the denominator goes up and becomes the numerator only if it is reduced to the power of the number it previously had!!