Find the slope of the tangent line

[cal.queen92]

Homework Statement

Find the slope of the tangent line to the curve:

sqrt(4x+2y) + sqrt(1xy) = 9.72

at the point (6,3)

Homework Equations

Derivative laws

The Attempt at a Solution

the slope of the tangent line to a curve is the Derivative of the function of the curve, so I need to find the derivative, right?

Now, to obtain the derivative, I feel as if I need to have the function as a function of y in terms of x. (f(x) = ... or y=x...). But every time I attempt to put the equation in that form I get stuck!

I tried squaring both sides of the equation, ending up with something ugly.

I also tried the following:

sqrt(4x+2y) + sqrt(1xy) = 9.72 --> sqrt(4x+2y) = 9.72 - sqrt(1xy)

and then squared both sides again and ended up with something ugly again.

Any Ideas?

Thanks!

 

[quantumdude]

the slope of the tangent line to a curve is the Derivative of the function of the curve, so I need to find the derivative, right?

Right!

Now, to obtain the derivative, I feel as if I need to have the function as a function of y in terms of x.

Wrong!

You don't have to have that, and in this case it isn't even desirable. Use implicit differentiation instead.

 

[cal.queen92]

OOOOkayy i completely forgot about implicit differentiation... I've tried it now and am not getting it right, The differentiating process is extremely long, should it be? I am getting lost in my work it is so long.

 

[quantumdude]

No, it shouldn't be long at all. Rewrite the relation as follows.

(4x+2y)^{1/2}+(xy)^{1/2}=9.72

You'll need to apply the Chain Rule to both terms on the left side. In addition to that you'll need the Product Rule for the second one. Don't forget that the derivative of the right side is zero.

 

[cal.queen92]

Yes, I used the chain rule and product rule and ended up with:

dy/dx = ((-xy^2 * sqrt(4x+2y)) -4)/ (2 + (yx^2 * sqrt(4x+2y))

and then when I filled in the values of the point i got an answer of:

(-54sqrt(30)-4)/(2+108sqrt(30)) which is wrong...

perhaps if i show you my first step:

((1/2)((4x+2y)^(-1/2))(4+2dy/dx)) + ((1/2)((xy)^ and I have just found my mistake...

Thank you very much! The answer is now right...