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

Solution:

Step:1

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

Step:2

Take the derivative term by term on the left side.

Step:2.1

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

Step:2.2

Find the derivative of $4x^3$.

Step:2.2.1

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

Step:2.2.2

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

Step:2.2.3

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

Step:2.3

Differentiate $x^{2}y$.

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

Step:2.4

Differentiate $-x{y}^3$.

Step:2.4.1

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

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: $-3x{y}^2\frac{dy}{dx}-y^3$.

Step:2.4.6

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

Step:2.5

Simplify the entire derivative expression.

Step:2.5.1

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

Step:2.5.2

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

Step:2.5.3

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

Step:3

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

Step:4

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

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: $(x^2-3x{y}^2)\frac{dy}{dx}=y^3-12x^2-2xy$.

Step:5.2

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

Step:5.3

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

Step:5.3.1

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

Step:5.3.2

Simplify the right side if necessary.

Step:5.3.3

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

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 $f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$.

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

4. Chain Rule: The derivative of a composite function $f(g(x))$ is $f^{\prime }(g(x))g^{\prime }(x)$.

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.