how do you find the derivative of a square root?

Question

for example x=sqrt(144-p). wouldn't you flip it and give it a negative 1/2 exponent? 1/2(144-p)^-1/2 and multiply it by -1...or am i dumb?

Answer 1

x=√(144-p)

The derivative of a square root is (1/2)*reciprocal, so (1/2)*(1/√...), or 1/(2√...)

Thus, the derivative of √(144-p) is:
1 / (2√(144-p))

First derivative is simply:
x=√(144 - p)
x²=144-p
p= - x²+144
dy/dx = -2x

Answer 2

dx/dp = (d/dp) sqrt(144-p)
= (d/dp) ( 144 - p )^(1/2)
= (-1)(1/2)( 144 - p)^(-1/2)
= -(1/2)(144 - p)^(-1/2).
You are perfectly correct, provided you are differentiating x with respect to p.

If you wanted dp/dx, then you would say:
x^2 = 144 - p
p = 144 - x^2
dp/dx = -2x.

Answer 3

if x = √u, then since √u = u^(1/2), then dx/du = (1/2)u^(-1/2) = 1/(2√u)

and if u = 144-p, du/dp = -1, so using the chain rule,

dx/dp = dx/du • du/dp = 1/[2√(144-p) • -1
dx/dp = -1/[2√(144-p)

Answer 4

Need more information! Is there another equation involved?