Question

what is the derivative of y=2247-1123*sqrt(4-x)


This is a distance formula and I am taking the derivative to get a velocity formula

Answer 1

I would convert the square root into a term with an exponent of 1/2

sqrt(4-x) -> (4-x)^1/2 and use the exponent rule for derivatives

Answer 2

Power rule followed by chain rule.

Answer 3

what would happen to the 2247?

Answer 4

the 1123 would become halved whenever you do power rule correct?

Answer 5

what is the derivative of a constant? ALWAYS zero
this is fundamental and should be part of the definition before knowing all the rules

Answer 6

if you doubt me, I can show you the proof that the derivative of a constant is zero
or you can check any introduction analysis book like Apostol's Calculus or Spivak Calculus

Answer 7

that's just odd because that would mean I was getting a negative number

Answer 8

f'(x) = -x^2 would be -2x correct?

Answer 9

right. using power rule

Answer 10

when i took the derivative i got
dy/dt=-561.5(4-x)^-1/2

Answer 11

remember that negative exponent gives you
\[x^{-1/n}=\frac{ 1 }{ x^{1/n} }\]

Answer 12

y'(x) = 1123/(2 sqrt(4-x))

Answer 13

why isn't the 1123 negative?

Answer 14

because there's a negative and a negative
\[f'(x) = \frac{ 1 }{ x }=\frac{ -1 }{ x^2 }\]

Answer 15

ohhhhh ok. I gotchya! I was hung up on that one part. I understand what's going on now. Thanks!!

Answer 16

d/dx(y) = d/dx(2247-1123 sqrt(4-x))
The derivative of y is y'(x):
y'(x) = d/dx(2247-1123 sqrt(4-x))
Differentiate the sum term by term and factor out constants:
y'(x) = d/dx(2247)-1123 d/dx(sqrt(4-x))
The derivative of 2247 is zero:
y'(x) = -1123 (d/dx(sqrt(4-x)))+0
Simplify the expression:
y'(x) = -1123 (d/dx(sqrt(4-x)))
Using the chain rule, d/dx(sqrt(4-x)) = ( dsqrt(u))/( du) ( du)/( dx), where u = 4-x and ( d)/( du)(sqrt(u)) = 1/(2 sqrt(u)):
y'(x) = -1123 (d/dx(4-x))/(2 sqrt(4-x))
Differentiate the sum term by term and factor out constants:
y'(x) = -(1123 d/dx(4)-d/dx(x))/(2 sqrt(4-x))
The derivative of 4 is zero:
y'(x) = -(1123 (-(d/dx(x))+0))/(2 sqrt(4-x))
Simplify the expression:
y'(x) = (1123 (d/dx(x)))/(2 sqrt(4-x))
The derivative of x is 1:
y'(x) = (1 1123)/(2 sqrt(4-x))

Answer 17

even wolframalpha confirms it LOL

Answer 18

haha well thank you!! now to finish the word problem!