Calc, Implicit Differentiation
posted by Michael .
Regard y as independent variable and x as dependant variable and find the slope of the tangent line to the curve (4x^2 + 2y2)^2  x^2y = 4588 at point (3,4).
Correct answer is 0.668827160493827
Here's what I did:
2(8x(dy/dx) + 4y) 2x(dx/dy)y + x^2 = 0
16x(dx/dy) + 8y  2x(dx/dy)y = x^2
16x(dx/dy)  2x(dx/dy)y = x^2  8y
(dx/dy)(16x 2xy) = x^2  8y
(dx/dy) = (x^2  8y)/(16x 2xy)
But this doesnt work.
You have
(4x^{2} + 2y^{2})^{2}  x^{2}y = 4588
It appears you started correctly and you know what you want, dy/dx, but I don't think you did the correct calculations.
It might help to think of this as
(f(x) + g(y))^{2}  h(x)*y = C
where f(x) = 4x^{2}, g(y)=2y^{2} and h(x)=x^{2}.
Then the differentiation should go
2(f(x) + g(y))*d/dx(f(x) + g(y))  h(x)*dy/dx h'(x)y = 0
Then f'(x)=8x, d/dx g(y)=g'(y)*dy/dx and h'(x)=2x
What I suggest, and this what I've done for real long expressions, is to write the symbols out, do the differentiation on the symbols, then substitute them into your equation.
Does this help?
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