using the chain rule to compute derivative given x:

[yanarains]

y=radical u; u=x^2 -2x+6
x=3

y=u^-1/2

dy/du= 1/u1/2 ; du/dx=2x-2

dy/dx=dy/du x du/dx = 1/u1/2 x 2x-2

=2x-1 / u1/2

then plug in 3 for the equation of : 3^2-2(3)+6 =9

then I plugged in x=3 and u=9 into the equation for dy/dx = 2(3)-1 / 9 1/2 = 5/3

Then answer is really 2/9. My question is when solving for dy/dx why is the answer x-1/u 1/2 instead of 2x-1/u 1/2? Could someone please explain this to me.

thank you in advance you for time. it is greatly appreciated.

 

[arthur ohlsten]

y=u^1/2
u=x^2-2x+6

take the derivative of y with respect to x
dy/dx = 1/2 u^(-1/2) du/dx
dy/dx = 1/[2u^1/2] [du/dx]

take derivative of u with respect to x
du/dx = 2x-2 substitute

dy/dx = 2[x-1]/[2u^1/2]
dy/dx = [x-1][u^1/2]

dy/dx = [3-1] /[9-6+6]^1/2
dy/dx= 2/3

=============================================
different approach
y=u^1/2
u=[x^2-2x+6] substitute

y=[x^2-2x+6]^1/2 take derivative
dy/dx = 1/2 [x^2-2x+6] ^-1/2 [2x-2]
dy/dx = [x-1]/[x^2-2x+6]
evaluate at x=3

dy/dx = 2/[9]^1/2
dy/dx= 2/3 or -2/3 answer

same answer: 2/3
Arthur