Find dy/dx by implicit differentiation.
arctan(2x^2y)=x+4xy^2
To find dy/dx by implicit differentiation, we will differentiate both sides of the equation with respect to x, treating y as an implicit function of x.
Given equation:
arctan(2x^2y) = x + 4xy^2
Differentiating both sides with respect to x, we get:
d/dx (arctan(2x^2y)) = d/dx (x + 4xy^2)
To find the derivative on the left-hand side, we need to use the chain rule. The chain rule states that if we have a composite function, like f(g(x)), the derivative can be found by multiplying the derivative of the outer function (f') by the derivative of the inner function (g').
Let's find the derivatives step by step:
Step 1: Differentiating the left-hand side
Using the chain rule, we have:
d/dx (arctan(2x^2y)) = 1/(1 + (2x^2y)^2) * d/dx(2x^2y)
Step 2: Differentiating the inner function
d/dx(2x^2y) = 2x^2(dy/dx) + 4xy
Step 3: Simplifying the left-hand side
Substituting the result from Step 2 into Step 1, we have:
1/(1 + (2x^2y)^2) * (2x^2(dy/dx) + 4xy) = x + 4xy^2
Step 4: Simplifying the equation
Multiplying through by (1 + (2x^2y)^2), we get:
2x^2(dy/dx) + 4xy = (1 + (2x^2y)^2)(x + 4xy^2)
Step 5: Expanding and rearranging the equation
Expanding the right-hand side and collecting like terms, we have:
2x^2(dy/dx) + 4xy = x + 4xy^2 + x(2x^2y)^2 + 4xy^2(2x^2y)^2
Simplifying further, we get:
2x^2(dy/dx) + 4xy = x + 4xy^2 + 4x^3y^2 + 32x^5y^5
Step 6: Rearranging the equation to isolate dy/dx
Subtracting (4x^3y^2 + 32x^5y^5) from both sides and rearranging, we have:
2x^2(dy/dx) = x - 4xy^2 - 4x^3y^2 - 32x^5y^5
Finally, dividing through by 2x^2 to solve for dy/dx, we get:
dy/dx = (x - 4xy^2 - 4x^3y^2 - 32x^5y^5) / (2x^2)
Therefore, the derivative of y with respect to x is given by:
dy/dx = (x - 4xy^2 - 4x^3y^2 - 32x^5y^5) / (2x^2)
To find dy/dx by implicit differentiation, you can follow these steps:
1. Start by differentiating both sides of the equation with respect to x, treating y as a function of x.
For the left side, you can use the chain rule because y is a composite function of x.
The derivative of arctan(u), where u is a function of x, is given by 1/(1+u^2) * du/dx.
Applying the chain rule, the left side can be differentiated as follows:
d(arctan(2x^2y))/dx = (1/(1+(2x^2y)^2)) * d(2x^2y)/dx
For the right side, you can simply differentiate each term with respect to x:
d(x)/dx + d(4xy^2)/dx = 1 + 4y^2 * dx/dx + 4x(dy^2)/dx
2. Simplify the differentiation on both sides of the equation.
Starting with the left side:
d(arctan(2x^2y))/dx = (1/(1+4x^4y^2)) * d(2x^2y)/dx
For the right side:
1 + 4y^2 + 4x(dy^2)/dx
3. Now, solve for dy/dx by moving all the terms involving dy/dx to one side of the equation and simplifying:
(1/(1+4x^4y^2)) * d(2x^2y)/dx - 4x(dy^2)/dx = 1 + 4y^2 - 1
Since (d(2x^2y)/dx) represents the derivative of 2x^2y with respect to x, we need to use the product rule for differentiation:
d(2x^2y)/dx = 2x^2 * dy/dx + 2y * d(x^2)/dx
= 2x^2 * dy/dx + 2y * 2x
So the equation becomes:
(1/(1+4x^4y^2)) * (2x^2 * dy/dx + 2y * 2x) - 4x(dy^2)/dx = 1 + 4y^2 - 1
4. Rearrange the terms and factor out dy/dx:
(2x^2/(1+4x^4y^2)) * dy/dx + (4xy^2/(1+4x^4y^2)) - (4x(dy^2)/dx) = 4y^2
Finally, solve for dy/dx:
dy/dx = (4y^2 - (4xy^2/(1+4x^4y^2)) + (4x(dy^2)/dx)) / (2x^2/(1+4x^4y^2))
This expression represents the derivative dy/dx in terms of x and y.
arctan(2x^2y)=x+4xy^2
Using the good old product and chain rules, we get
1/(1+(2x^2y)^2) * (4xy + 2x^2y') = 1 + 4y^2 + 8xyy'
Now just collect terms and solve for y':
16x^4y^4 + 4x^4y^2 + 4y^2 - 4xy + 1
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2x(16x^4y^3 - x + 4y)