The problem::
Use implicit differentiation to find the slope and the equation of the tangent line to the curve: sqrt x + sqrt y =9 at the point (16,25)
I dont know how to get the differentiation of sqrt of x and sqrt of y, but i do know that the 9 becomes a 0. Thanks, any help is appreciated.
2006-10-26 19:21:42 · 5 answers · asked by Slevin Kelevra 2 in Science & Mathematics ➔ Mathematics
Well, the slope is just the differentation, so to differentiate the equation of the curve, we must take the derivative of that equation:
By converting the sqrt into 1/2 power, we can rewrite the equation as:
x^(1/2) + y^(1/2) = 9
Now, to find the derivative in respect to x (dy/dx):
1/2x^(-1/2) + [1/2y^(-1/2) * (dy/dx)] = 0. (Since we're taking the derivative of x, we have to leave dy/dx multiplied to the derivative of the y variable.)
Next, we have to isolate dy/dx on one side of the equation and everything else on the other side.
1/2y^(-1/2)*(dy/dx) = -1/2x^(-1/2)
Now, dividing the equation by 1/2y^(-1/2) on both sides (and converting x^(-1/2) to 1/sqrt(x) and similar to y):
dy/dx = -sqrt(y)/sqrt(x).
Substituting in (16,25) gives us -5/4, the slope.
By using the point slope formula, we can plug in the numbers:
y-25=-5/4(x-16)
y-25=-5/4x+20
y=-5/4x+45
The above equation is the equation of the tangent line to the curve.
2006-10-26 19:30:55 · answer #1 · answered by Gimmip 2 · 1⤊ 0⤋
The square root of x is the same as x to the power of a half. Subtract the x to a half to the other side to isolate the square root of y. To get rid of the y's square root, square both sides. This will give you y=(9-x to the half)squared. Now you don't have to worry about implicit differentiation, and you can solve it by finding the derivative of the right side. Good luck.
2006-10-27 02:42:15 · answer #2 · answered by Anonymous · 0⤊ 0⤋
Take the derivative of every part of the function with respect to x, noting in particular that y here is a function, not a constant, so its derivative will be denoted dy/dx.
2006-10-27 02:25:58 · answer #3 · answered by topher8128 2 · 0⤊ 0⤋
sqrt x + sqrt y =9
(1/2)dx/sqrt(x) + (1/2)dy)/sqrt(y) = 0
(1/2)dx/sqrt(x) = -(1/2)dy)/sqrt(y)
dy/dx = -sqrt(x)/sqrt(y)
@ (16,25)
dy/dx = -sqrt(16)/sqrt(25)
dy/dx = -4/5
2006-10-27 02:44:31 · answer #4 · answered by Helmut 7 · 0⤊ 1⤋
wow, what kinda math is this. i feel really dumb now, i thought i was good at math and was excited to answer your question until i read it. sorry i wish i could help
2006-10-27 02:25:10 · answer #5 · answered by Anonymous · 0⤊ 0⤋