Derivatives????

Question

What is the derivative of the square root of x?

√(x)

Answer 1

sqrt(x) is the same as x^(½)

The derivative of this is ½ * x^(-½) =

1/(2x^½) =

1/(2√x) =

(√x)/(2x) { Because it's not nice to have radicals in the denominator }

Answer 2

Each person understood it, but did not explain it.
The problem involves properties of exponents.

sqrt(x) = x^(1/2)
der(x)^2 = (1/2)(x^(-1/2))
= 1/2 /(x^1/2)
= 1/2(x^1/2)

Answer 3

d/dx u^n = n * u^(n - 1)

√(x) = x^(1/2)

d/dx x^(1/2) = 1/2 * x^(1/2 - 1)
= 1/2 * x^(-1/2)
= 1/2 * 1/x^(1/2)
= 1/[2√(x)] <== answer

Answer 4

d(√x)/dx
d(x^(1/2))/dx
1/2 x^(-1/2)
1/(2√x)

Or directly:

d(√x)/dx
[h→0]lim (√(x+h) - √x)/h
[h→0]lim (x+h - x)/(h(√(x+h) + √x))
[h→0]lim h/(h(√(x+h) + √x))
[h→0]lim 1/(√(x+h) + √x)
1/(√(x+0) + √x)
1/(2√x)

Q.E.D.

Answer 5

1/2 x ^ (-1/2) or .5x^(-.5)