Using 4x^2 - 9y^2 = 36
Find y by implicit differentiation and then solve the equation explicitly for y and differentiate to get y' in terms of x.
I am sure you wanted y'
8x - 18y dy/dx = 0
-18y dy/dx = -8x
dy/dx = 8x/(18y) = 4x/(9y)
-9y^2 = 36 - 4x^2
y^2 = (4x^2 - 36)/9
y = (4x^2 - 36)^(1/2) /3
dy/dx = (1/6) (4x^2 - 36)^(-1/2) (8x)
= (4/3)(x)(4x^2 - 36)^(-1/2)
Oh boy, implicit differentiation, my favorite! Let's dive in and have some fun!
First, we need to differentiate both sides of the equation with respect to x. So, get ready for some prime fun! Here we go!
Differentiating the left side, we'll use the power rule for x:
d(4x^2)/dx = 8x
Now differentiating the right side:
d(36)/dx = 0 (since 36 is just a constant)
Now time to differentiate the fun part, the y terms! We'll use the chain rule here:
d(-9y^2)/dx = -18y * dy/dx
Now let's put it all together:
8x - 18y * dy/dx = 0
To solve for dy/dx, we just need to get it all by itself:
-18y * dy/dx = -8x
Now we divide both sides by -18y:
dy/dx = (-8x) / (-18y)
Simplifying a bit, we get:
dy/dx = 4x / 9y
And there you have it! We found y' (dy/dx) in terms of x. It may not be the most exciting answer, but hey, at least it wasn't a clown show, right?
To find y by implicit differentiation, we differentiate both sides of the equation with respect to x:
Differentiating 4x^2 - 9y^2 = 36 with respect to x:
d/dx(4x^2) - d/dx(9y^2) = d/dx(36)
Using the power rule for differentiation:
8x - 18y(dy/dx) = 0
Now, let's solve for y explicitly. Rearranging the equation:
18y(dy/dx) = 8x
dy/dx = (8x) / (18y)
To differentiate and express y' in terms of x, let's solve the equation for y explicitly:
18y(dy/dx) = 8x
dy/dx = (8x) / (18y)
Now, let's solve the equation for y explicitly:
18y(dy/dx) = 8x
(dy/dx) = (8x) / (18y)
Multiply both sides by (18y):
18y(dy/dx) = 8x
Divide both sides by 18:
dy/dx = (8/18)(x/y)
Since dy/dx = y', we have:
y' = (8/18)(x/y)
To find y by implicit differentiation, we need to differentiate both sides of the equation with respect to x. Let's start by differentiating each term separately using the chain rule:
For the left side, we have:
d/dx (4x^2 - 9y^2) = d/dx (4x^2) - d/dx (9y^2)
Differentiating 4x^2 with respect to x gives us:
d/dx (4x^2) = 8x
For the second term, we need to apply the chain rule. Let's set v = y^2:
d/dx (9v) = d/dv (9v) * dv/dx
Differentiating 9v with respect to v gives us:
d/dv (9v) = 9
Differentiating v = y^2 with respect to x gives us:
dv/dx = d/dx (y^2) = 2y * dy/dx
Now, let's substitute all the derivatives back into the equation:
8x - 9 * 2y * dy/dx = 0
Simplifying the equation gives us:
8x - 18y * dy/dx = 0
In order to solve the equation explicitly for y, we can rearrange the terms:
18y * dy/dx = 8x
dy/dx = 8x / 18y
To differentiate to get y' in terms of x, we can rewrite dy/dx as y' and solve for y':
y' = 8x / 18y
Now we have the equation for y' in terms of x, obtained by implicit differentiation.