Find the derivative of the implicit function (X^2+Y^2)^3=8X^2Y^2.
To find the derivative of the implicit function, we can first rewrite it in terms of Y explicitly.
(X^2 + Y^2)^3 = 8X^2Y^2
Expand the left side:
(X^6 + 3X^4Y^2 + 3X^2Y^4 + Y^6) = 8X^2Y^2
Now, differentiate both sides of the equation with respect to X:
6X^5 + 12X^3Y^2 + 6XY^4 + 6Y^5 * (dY/dX) = 16XY^2 + 16X^2Y * (dY/dX)
Move all terms involving (dY/dX) to one side:
6X^5 + 12X^3Y^2 + 6XY^4 - 16XY^2 - 16X^2Y = -6Y^5 * (dY/dX) + 6Y^2 * (dY/dX)
Now, factor out the common factor of (dY/dX):
6X^5 + 12X^3Y^2 + 6XY^4 - 16XY^2 - 16X^2Y = (6Y^2 - 6Y^5) * (dY/dX)
Finally, divide both sides by (6Y^2 - 6Y^5) to solve for (dY/dX):
(dY/dX) = (6X^5 + 12X^3Y^2 + 6XY^4 - 16XY^2 - 16X^2Y) / (6Y^2 - 6Y^5)
To find the derivative of the implicit function, we need to differentiate both sides of the equation with respect to the variable.
Let's start with the left-hand side of the equation: (X^2 + Y^2)^3.
Using the chain rule, we can differentiate this term as follows:
d/dx [(X^2 + Y^2)^3] = 3(X^2 + Y^2)^2 * d/dx (X^2 + Y^2)
To differentiate X^2 with respect to x, we treat Y^2 as a constant. Thus, we get:
d/dx (X^2) = 2X * dX/dx = 2X * 1 = 2X
Similarly, differentiating Y^2 with respect to x, we treat X^2 as a constant. Since Y^2 does not contain x, its derivative is zero.
Now let's move to the right-hand side of the equation: 8X^2Y^2.
Differentiating 8X^2Y^2 with respect to x requires the product rule. We evaluate the derivative of X^2 and Y^2 separately, then multiply them by the original terms.
For Y^2, since it does not contain x, its derivative is zero.
For X^2, we get:
d/dx (X^2) = 2X * dX/dx = 2X * 1 = 2X
Now, let's put everything together:
3(X^2 + Y^2)^2 * (2X) = 8X^2 * Y^2
Simplifying further:
6X(X^2 + Y^2)^2 = 8X^2 * Y^2
This is the derivative of the implicit function (X^2 + Y^2)^3 = 8X^2Y^2 with respect to x.
To find the derivative of the implicit function, we will use the implicit differentiation method. Here's how you can do it step by step:
Step 1: Start by differentiating both sides of the equation with respect to the variable (usually x or y). In this case, we'll differentiate with respect to x.
Step 2: For the left-hand side (LHS) of the equation, we need to apply the chain rule. Let's break it down:
The derivative of (X^2+Y^2)^3 with respect to x is:
d/dx [(X^2+Y^2)^3] = 3(X^2+Y^2)^2 * 2X * dx/dx + 3(X^2+Y^2)^2 * 2Y * dy/dx
Step 3: For the right-hand side (RHS) of the equation, we can differentiate each term individually. Let's break it down:
The derivative of 8X^2Y^2 with respect to x is:
d/dx [8X^2Y^2] = 16XY^2 + 8X^2 * dY/dx
Step 4: Now, equate the derivatives obtained from LHS and RHS:
3(X^2+Y^2)^2 * 2X * dx/dx + 3(X^2+Y^2)^2 * 2Y * dy/dx = 16XY^2 + 8X^2 * dY/dx
Step 5: Now, we can isolate dy/dx to find the derivative with respect to x:
3(X^2+Y^2)^2 * 2Y * dy/dx - 8X^2 * dY/dx = 16XY^2 - 3(X^2+Y^2)^2 * 2X
dy/dx * [3(X^2+Y^2)^2 * 2Y - 8X^2] = 16XY^2 - 3(X^2+Y^2)^2 * 2X
dy/dx = (16XY^2 - 3(X^2+Y^2)^2 * 2X) / [3(X^2+Y^2)^2 * 2Y - 8X^2]
And there you have it! The derivative of the implicit function can be found using this formula.