What is the derivitive of 5 times the square root of x?

Answer 1

How do you find the derivative of x^c ?
You take c and put it in front as a coefficient, then drop the exponent by 1

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

square root is the same as x^(1/2)
Our c is 1/2

Also, the 5 in front is a factor and it can be taken outside.

d(5f) = 5*d(f)

so derivative of 5x^(1/2)
= 5 times the derivative of x^(1/2)
= 5 times (1/2) x^(1/2 - 1)
= (5/2) x^(-1/2)

A negative exponent means an inverse
(e.g., x^-1 = 1/x )

= (5/2) / x^(1/2)
= 5 / 2√x
= 5 over 2 times the square root of x.

In general one does not leave a root in the denominator. We can always multiply by 1 without changing the value (√x / √x = 1)
multiply the numerator by √x and the denominator by √x:

5√x / (2 √x √x) = 5√x / 4 x