Find dy/dx 4x^3+x^2y-xy^3=1

Solution:

Step:1

Apply the derivative operator to both sides of the equation: $\frac{d}{dx}(4x^3 + x^2y - xy^3) = \frac{d}{dx}(1)$.

Step:2

Take the derivative term by term on the left side.

Step:2.1

Use the Sum Rule to separate the derivatives: $\frac{d}{dx}(4x^3) + \frac{d}{dx}(x^2y) + \frac{d}{dx}(-xy^3)$.

Step:2.2

Find the derivative of $4x^3$.

Step:2.2.1

Extract the constant $4$ and differentiate $x^3$: $4\frac{d}{dx}(x^3) + \frac{d}{dx}(x^2y) + \frac{d}{dx}(-xy^3)$.

Step:2.2.2

Apply the Power Rule to $x^3$: $4(3x^2) + \frac{d}{dx}(x^2y) + \frac{d}{dx}(-xy^3)$.

Step:2.2.3

Simplify the result: $12x^2 + \frac{d}{dx}(x^2y) + \frac{d}{dx}(-xy^3)$.

Step:2.3

Differentiate $x^2y$.

Step:2.3.1

Use the Product Rule: $x^2\frac{d}{dx}(y) + y\frac{d}{dx}(x^2)$.

Step:2.3.2

Recognize $\frac{d}{dx}(y)$ as $\frac{dy}{dx}$.

Step:2.3.3

Apply the Power Rule to $x^2$: $2x$.

Step:2.3.4

Combine the terms: $12x^2 + x^2\frac{dy}{dx} + 2xy + \frac{d}{dx}(-xy^3)$.

Step:2.4

Differentiate $-xy^3$.

Step:2.4.1

Factor out the constant $-1$: $-1\frac{d}{dx}(xy^3)$.

Step:2.4.2

Apply the Product Rule: $-x\frac{d}{dx}(y^3) - y^3\frac{d}{dx}(x)$.

Step:2.4.3

Use the Chain Rule for $y^3$.

Step:2.4.3.1

Substitute $u$ for $y$: $-x(3u^2\frac{d}{dx}(y)) - y^3$.

Step:2.4.3.2

Apply the Power Rule to $u^3$: $3u^2$.

Step:2.4.3.3

Replace $u$ with $y$: $-x(3y^2\frac{dy}{dx}) - y^3$.

Step:2.4.4

Recognize $\frac{d}{dx}(x)$ as $1$.

Step:2.4.5

Simplify the expression: $-3xy^2\frac{dy}{dx} - y^3$.

Step:2.4.6

Combine all terms: $12x^2 + x^2\frac{dy}{dx} + 2xy - 3xy^2\frac{dy}{dx} - y^3$.

Step:2.5

Simplify the entire derivative expression.

Step:2.5.1

Combine like terms: $12x^2 + x^2\frac{dy}{dx} + 2xy - 3xy^2\frac{dy}{dx} - y^3$.

Step:2.5.2

Reorder terms for clarity: $12x^2 + x^2\frac{dy}{dx} + 2xy - y^3 - 3xy^2\frac{dy}{dx}$.

Step:2.5.3

Final simplified left side: $12x^2 + (x^2 - 3xy^2)\frac{dy}{dx} + 2xy - y^3$.

Step:3

The derivative of a constant is zero: $\frac{d}{dx}(1) = 0$.

Step:4

Combine the derivatives to form the differential equation: $12x^2 + (x^2 - 3xy^2)\frac{dy}{dx} + 2xy - y^3 = 0$.

Step:5

Isolate $\frac{dy}{dx}$ to solve for the derivative.

Step:5.1

Move all terms not containing $\frac{dy}{dx}$ to the other side: $(x^2 - 3xy^2)\frac{dy}{dx} = y^3 - 12x^2 - 2xy$.

Step:5.2

Factor out $\frac{dy}{dx}$ from the left side: $\frac{dy}{dx}(x^2 - 3xy^2) = y^3 - 12x^2 - 2xy$.

Step:5.3

Divide both sides by $(x^2 - 3xy^2)$ to solve for $\frac{dy}{dx}$.

Step:5.3.1

Perform the division: $\frac{dy}{dx} = \frac{y^3 - 12x^2 - 2xy}{x^2 - 3xy^2}$.

Step:5.3.2

Simplify the right side if necessary.

Step:5.3.3

Final expression for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{y^3 - 12x^2 - 2xy}{x^2 - 3xy^2}$.

Step:6

The derivative $\frac{dy}{dx}$ is now expressed in terms of $x$ and $y$.

Knowledge Notes:

1. Sum Rule: The derivative of a sum of functions is the sum of their derivatives.

2. Product Rule: The derivative of a product of two functions $f(x)g(x)$ is $f'(x)g(x) + f(x)g'(x)$.

3. Power Rule: The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.

4. Chain Rule: The derivative of a composite function $f(g(x))$ is $f'(g(x))g'(x)$.

5. Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.

6. Implicit Differentiation: When a function is not given explicitly as $y=f(x)$, but instead is given in a form that relates $x$ and $y$, we differentiate both sides of the equation with respect to $x$ and solve for $\frac{dy}{dx}$.

7. Simplification: After taking the derivative, it is often necessary to simplify the expression by combining like terms, factoring, and reducing fractions.