Find dy/dx xy^3-7y+y^2-4=0

Solution:

Step:1

Apply differentiation to both sides of the equation with respect to $x$: $\frac{d}{dx}(x{y}^3-7y+y^2-4)=\frac{d}{dx}(0)$.

Step:2

Differentiate each term on the left side individually.

Step:2.1

Use the Sum Rule for differentiation: $\frac{d}{dx}(x{y}^3)+\frac{d}{dx}(-7y)+\frac{d}{dx}(y^2)+\frac{d}{dx}(-4)$.

Step:2.2

Differentiate the term $x{y}^3$.

Step:2.2.1

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

Step:2.2.2

Use the Chain Rule for $y^3$.

Step:2.2.2.1

Let $u_1=y$ and differentiate $u_1^3$: $x(3u_1^2\frac{d}{dx}(y))+y^3$.

Step:2.2.2.2

Apply the Power Rule to $u_1^3$: $x(3y^2\frac{d}{dx}(y))+y^3$.

Step:2.2.2.3

Substitute $u_1$ back with $y$: $x(3y^2\frac{dy}{dx})+y^3$.

Step:2.2.3

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

Step:2.2.4

Differentiate $x$ using the Power Rule: $x(3y^2\frac{dy}{dx})+y^3(1)$.

Step:2.2.5

Combine constants with variables: $3x{y}^2\frac{dy}{dx}+y^3$.

Step:2.2.6

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

Step:2.3

Differentiate the term $-7y$.

Step:2.3.1

Multiply the derivative of $y$ by the constant $-7$: $-7\frac{dy}{dx}$.

Step:2.3.2

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

Step:2.4

Differentiate the term $y^2$.

Step:2.4.1

Use the Chain Rule for $y^2$.

Step:2.4.1.1

Let $u_2=y$ and differentiate $u_2^2$: $2u_2\frac{d}{dx}(y)$.

Step:2.4.1.2

Apply the Power Rule to $u_2^2$: $2y\frac{dy}{dx}$.

Step:2.4.1.3

Substitute $u_2$ back with $y$: $2y\frac{dy}{dx}$.

Step:2.4.2

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

Step:2.5

The derivative of a constant $-4$ is $0$.

Step:2.6

Combine all differentiated terms: $3x{y}^2\frac{dy}{dx}+y^3-7\frac{dy}{dx}+2y\frac{dy}{dx}$.

Step:3

The derivative of the right side $0$ is $0$.

Step:4

Set the differentiated left side equal to the differentiated right side: $3x{y}^2\frac{dy}{dx}+y^3-7\frac{dy}{dx}+2y\frac{dy}{dx}=0$.

Step:5

Solve for $\frac{dy}{dx}$.

Step:5.1

Isolate terms with $\frac{dy}{dx}$ on one side: $3x{y}^2\frac{dy}{dx}-7\frac{dy}{dx}+2y\frac{dy}{dx}=-y^3$.

Step:5.2

Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(3x{y}^2-7+2y)=-y^3$.

Step:5.3

Divide both sides by $(3x{y}^2-7+2y)$ to solve for $\frac{dy}{dx}$: $\frac{dy}{dx}=-\frac{y^3}{3x{y}^2-7+2y}$.

Step:6

Express the final result: $\frac{dy}{dx}=-\frac{y^3}{3x{y}^2-7+2y}$.

Knowledge Notes:

1. Differentiation: The process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable.

2. Sum Rule: The derivative of a sum of functions is the sum of the derivatives of each function.

3. Product Rule: The derivative of the product of two functions is given by $d(uv)/dx=u(dv/dx)+v(du/dx)$.

4. Chain Rule: The derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

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

6. Constant Rule: The derivative of a constant is $0$.

7. Implicit Differentiation: When differentiating an equation involving two variables, treat the non-differentiation variable as a function of the differentiation variable and apply the chain rule as necessary.