 Find dy/dx by implicit differentiation
 x2y +3xy3 −x =3 
6. x3y2 −5x2y +x =1 
7. 1 √x + 1 √y =1 
8. x2 = x +y x −y 9. sin(x2y2) = x 10. cos(xy2) = y 11. tan3(xy2 +y)= x 12. xy3 1+secy =1+y4 

Let's solve the given problems by finding \(\frac{dy}{dx}\) using implicit differentiation.

### 1. \(x^2y + 3xy^3 - x = 3\)
Differentiate both sides with respect to \(x\):
\[
\frac{d}{dx}(x^2y) + \frac{d}{dx}(3xy^3) - \frac{d}{dx}(x) = \frac{d}{dx}(3)
\]
Apply the product rule:
\[
2xy + x^2\frac{dy}{dx} + 3y^3 + 9xy^2\frac{dy}{dx} - 1 = 0
\]
Group the terms involving \(\frac{dy}{dx}\):
\[
(x^2 + 9xy^2)\frac{dy}{dx} = 1 - 2xy - 3y^3
\]
Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{1 - 2xy - 3y^3}{x^2 + 9xy^2}
\]

### 2. \(x^3y^2 - 5x^2y + x = 1\)
Differentiate both sides with respect to \(x\):
\[
\frac{d}{dx}(x^3y^2) - \frac{d}{dx}(5x^2y) + \frac{d}{dx}(x) = \frac{d}{dx}(1)
\]
Apply the product rule:
\[
3x^2y^2 + 2x^3y\frac{dy}{dx} - 10xy - 5x^2\frac{dy}{dx} + 1 = 0
\]
Group the terms involving \(\frac{dy}{dx}\):
\[
(2x^3y - 5x^2)\frac{dy}{dx} = -3x^2y^2 + 10xy - 1
\]
Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{-3x^2y^2 + 10xy - 1}{2x^3y - 5x^2}
\]

### 3. \(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}} = 1\)
Rewrite the equation as:
\[
x^{-\frac{1}{2}} + y^{-\frac{1}{2}}\frac{dy}{dx} = 0
\]
Differentiate both sides with respect to \(x\):
\[
-\frac{1}{2}x^{-\frac{3}{2}} + \left(-\frac{1}{2}y^{-\frac{3}{2}}\right)\frac{dy}{dx} = 0
\]
Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{\sqrt{y}}{\sqrt{x}}
\]

### 4. \(x^2 = \frac{x + y}{x - y}\)
Differentiate both sides with respect to \(x\):
\[
2x = \frac{(x-y)\left(\frac{d}{dx}(x+y)\right) - (x+y)\left(\frac{d}{dx}(x-y)\right)}{(x-y)^2}
\]
Simplify and solve for \(\frac{dy}{dx}\):
\[
2x(x - y)^2 = (x - y) - (x + y)\frac{dy}{dx}
\]
\[
\frac{dy}{dx} = \frac{(x-y)(2x - 1)}{(x+y)}
\]

### 5. \(\sin(x^2y^2) = x\)
Differentiate both sides with respect to \(x\):
\[
\cos(x^2y^2) \cdot \left(2xy^2\frac{dy}{dx} + 2x^2y\right) = 1
\]
Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{1 - 2x^2y\cos(x^2y^2)}{2xy^2\cos(x^2y^2)}
\]

### 6. \(\cos(xy^2) = y\)
Differentiate both sides with respect to \(x\):
\[
-\sin(xy^2) \cdot \left(y^2 + 2xy\frac{dy}{dx}\right) = \frac{dy}{dx}
\]
Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{-y^2\sin(xy^2)}{1 + 2x\sin(xy^2)}
\]

### 7. \(\tan^3(xy^2 + y) = x\)
Differentiate both sides with respect to \(x\):
\[
3\tan^2(xy^2 + y) \cdot \sec^2(xy^2 + y) \cdot (y^2 + 2xy\frac{dy}{dx} + \frac{dy}{dx}) = 1
\]
Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{1 - 3\tan^2(xy^2 + y) \cdot \sec^2(xy^2 + y) \cdot y^2}{6xy\tan^2(xy^2 + y)\cdot \sec^2(xy^2 + y)}
\]

### 8. \(\frac{xy^3}{1 + \sec y} = 1 + y^4\)
Differentiate both sides with respect to \(x\):
\[
\frac{y^3 + 3xy^2\frac{dy}{dx}}{1 + \sec y} - \frac{xy^3\sec y \tan y\frac{dy}{dx}}{(1 + \sec y)^2} = 4y^3\frac{dy}{dx}
\]
Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{(y^3 + 4xy^3\sec y\tan y) - 1}{3xy^2(1 + \sec y) + 4y^3}
\]

Would you like more details or have any questions?

Here are 8 related problems you can try:
1. Differentiate implicitly \(x^3 + y^3 = 3xy\).
2. Solve \(e^{xy} = x + y\) using implicit differentiation.
3. Find \(\frac{dy}{dx}\) for \(xy = \ln(x + y)\).
4. Differentiate \(x^2 + y^2 + 2xy = 7\) implicitly.
5. Solve \(\sin(xy) = x + y\) using implicit differentiation.
6. Differentiate \(x^2y + y^2 = x + y\).
7. Find \(\frac{dy}{dx}\) for \(x^3y + xy^3 = 1\).
8. Solve \(x^2 + y^2 = \cos(xy)\) implicitly.

**Tip:** When differentiating implicitly, always remember to apply the product and chain rules appropriately to ensure you account for all terms involving \(y\).

Learn how to solve implicit differentiation problems step-by-step with detailed explanations and examples. This page covers equations involving radicals and advanced techniques in calculus. Suitable for advanced high school and college students.

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