Find dy/dx xy^3-7y+y^2-4=0
The problem is asking for the derivation of the given equation with respect to x. It involves finding the derivative dy/dx of the implicit function defined by the equation xy^3 - 7y + y^2 - 4 = 0, where x and y are variables. The terms within the equation are composed of both x and y, making it an implicit differentiation problem rather than explicit. The goal is to use differentiation rules, including the product rule and chain rule, to differentiate each term with respect to x, while also applying the implicit differentiation principle which involves taking the derivative of y with respect to x as dy/dx where necessary.
$x y^{3} - 7 y + y^{2} - 4 = 0$
Answer
Solution:
Step:1
Apply differentiation to both sides of the equation with respect to $x$: $\frac{d}{dx}(xy^3 - 7y + y^2 - 4) = \frac{d}{dx}(0)$.
Step:2
Differentiate each term on the left side individually.
Step:2.1
Use the Sum Rule for differentiation: $\frac{d}{dx}(xy^3) + \frac{d}{dx}(-7y) + \frac{d}{dx}(y^2) + \frac{d}{dx}(-4)$.
Step:2.2
Differentiate the term $xy^3$.
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: $3xy^2 \frac{dy}{dx} + y^3$.
Step:2.2.6
Simplify the expression: $3xy^2 \frac{dy}{dx} + y^3$.
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: $3xy^2 \frac{dy}{dx} + y^3 - 7 \frac{dy}{dx} + 2y \frac{dy}{dx}$.
Step:3
The derivative of the right side $0$ is $0$.
Step:4
Set the differentiated left side equal to the differentiated right side: $3xy^2 \frac{dy}{dx} + y^3 - 7 \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$.
Step:5
Solve for $\frac{dy}{dx}$.
Step:5.1
Isolate terms with $\frac{dy}{dx}$ on one side: $3xy^2 \frac{dy}{dx} - 7 \frac{dy}{dx} + 2y \frac{dy}{dx} = -y^3$.
Step:5.2
Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(3xy^2 - 7 + 2y) = -y^3$.
Step:5.3
Divide both sides by $(3xy^2 - 7 + 2y)$ to solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{y^3}{3xy^2 - 7 + 2y}$.
Step:6
Express the final result: $\frac{dy}{dx} = -\frac{y^3}{3xy^2 - 7 + 2y}$.
Knowledge Notes:
Differentiation: The process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable.
Sum Rule: The derivative of a sum of functions is the sum of the derivatives of each function.
Product Rule: The derivative of the product of two functions is given by $d(uv)/dx = u(dv/dx) + v(du/dx)$.
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.
Power Rule: The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
Constant Rule: The derivative of a constant is $0$.
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.
