Alright so implicit differentiation is just not working out for me.
Use implicit differentiation to find the slope of the tangent line to the curve at point (4,1).
y / (x-2y) = x^3 + 4
Tried quotient rule to get the derivative of the left side, then got derivative of right side, and ultimately I got y + 3x^2 (x-2y)^2 / x-2y-2y
This is clearly not correct since the bottom would turn into zero if I plugged in x and y. Sigh...
It will be easier if you multiply it out and combine terms.
9y = x^4 +4x -2x^3 y
Differentiate both sides with respect to x implicitly, treating y as a function of x.
9 dy/dx = 4x^3 + 4 -2x^3 dy/dx
-6x^2 y
dy/dx(9 + 2x^3) = 4x^3 + 4 - 6x^2 y
dy/dx = [4x^3 -6x^2 y +4]/(9 + 2x^3)
Plug in x=4 and y=1. You should not get a zero deniminator.
Don't worry, I can help you with implicit differentiation!
To find the slope of the tangent line to the curve at point (4, 1), we first need to differentiate both sides of the given equation with respect to x. Let's go step by step:
1. Start by differentiating the left side of the equation. For this, we need to use the chain rule since we have y as a function of x. To differentiate y / (x - 2y) with respect to x, we need to differentiate y with respect to x and apply the quotient rule.
The derivative of y with respect to x is dy/dx. And for the quotient rule:
(d/dx)(y / (x - 2y)) = [(x - 2y)(dy/dx) - y(-1)] / (x - 2y)^2
2. Next, differentiate the right side of the equation. The derivative of x^3 + 4 with respect to x is simply 3x^2.
3. Now that we have the derivatives of both sides of the equation, we can equate them to find the derivative of y with respect to x. So, the equation becomes:
[(x - 2y)(dy/dx) - y(-1)] / (x - 2y)^2 = 3x^2
4. We want to find the slope of the tangent line at the point (4, 1), so substitute x = 4 and y = 1 into the equation:
[(4 - 2(1))(dy/dx) - 1(-1)] / (4 - 2(1))^2 = 3(4)^2
Simplifying:
[2(dy/dx) + 1] / 2^2 = 48
[2(dy/dx) + 1] / 4 = 48
2(dy/dx) + 1 = 48 * 4
2(dy/dx) + 1 = 192
2(dy/dx) = 192 - 1
2(dy/dx) = 191
dy/dx = 191/2
So, the slope of the tangent line to the curve at point (4, 1) is dy/dx = 191/2.
It seems like there might have been an error in simplifying your derivative expression. Double-check your calculations to make sure you didn't miss any terms or signs.