Find dy/dx 3xy+y^2=4x+y
The given question is asking for the derivative of a function with respect to x, symbolized as dy/dx, which means you need to find how the function's rate of change in y corresponds to a small change in x. The function is implicitly defined by the equation 3xy + y^2 = 4x + y. Instead of being given as y = f(x), the relationship between x and y is expressed in a form that intertwines both variables. The task here is to differentiate this equation with respect to x, taking into account the derivative of y with respect to x whenever y is involved, likely using implicit differentiation methods.
$3 x y + y^{2} = 4 x + y$
Answer
Solution:
Step 1
Take the derivative of each term in the equation $3xy + y^2 = 4x + y$ with respect to $x$.
$$\frac{d}{dx}(3xy + y^2) = \frac{d}{dx}(4x + y)$$
Step 2
Differentiate the left-hand side of the equation.
Step 2.1
Apply the Sum Rule to separate the derivatives: $\frac{d}{dx}(3xy) + \frac{d}{dx}(y^2)$.
Step 2.2
Find the derivative of $3xy$.
Step 2.2.1
Extract the constant 3: $3\frac{d}{dx}(xy)$.
Step 2.2.2
Use the Product Rule: $\frac{d}{dx}(f(x)g(x)) = f(x)\frac{d}{dx}g(x) + g(x)\frac{d}{dx}f(x)$, where $f(x) = x$ and $g(x) = y$.
$$3(x\frac{d}{dx}y + y\frac{d}{dx}x)$$
Step 2.2.3
Substitute $\frac{d}{dx}y$ with $dy/dx$.
$$3(xy + y)$$
Step 2.2.4
Apply the Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n = 1$.
$$3(xy + y)$$
Step 2.2.5
Simplify by multiplying $y$ by 1.
$$3(xy + y)$$
Step 2.3
Find the derivative of $y^2$.
Step 2.3.1
Use the Chain Rule: $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$, where $f(x) = x^2$ and $g(x) = y$.
Step 2.3.1.1
Set $u = y$ and differentiate $u^2$ with respect to $u$.
$$3(xy + y) + 2u\frac{dy}{dx}$$
Step 2.3.1.2
Apply the Power Rule: $\frac{d}{du}(u^n) = nu^{n-1}$, where $n = 2$.
$$3(xy + y) + 2y\frac{dy}{dx}$$
Step 2.3.1.3
Replace $u$ with $y$.
$$3(xy + y) + 2y\frac{dy}{dx}$$
Step 2.3.2
Express $\frac{d}{dx}y$ as $dy/dx$.
$$3(xy + y) + 2y(dy/dx)$$
Step 2.4
Combine terms.
Step 2.4.1
Distribute $3$ across $xy$ and $y$.
$$3xy + 3y + 2y(dy/dx)$$
Step 2.4.2
Arrange the terms in order.
$$3xy + 2y(dy/dx) + 3y$$
Step 3
Differentiate the right-hand side of the equation.
Step 3.1
Apply the Sum Rule: $\frac{d}{dx}(4x) + \frac{d}{dx}(y)$.
Step 3.2
Find the derivative of $4x$.
Step 3.2.1
Extract the constant 4: $4\frac{d}{dx}(x)$.
Step 3.2.2
Apply the Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n = 1$.
$$4 + \frac{dy}{dx}$$
Step 3.2.3
Simplify by multiplying $4$ by 1.
$$4 + \frac{dy}{dx}$$
Step 3.3
Express $\frac{d}{dx}y$ as $dy/dx$.
$$4 + dy/dx$$
Step 4
Combine the derivatives from both sides of the equation.
$$3xy + 2y(dy/dx) + 3y = 4 + dy/dx$$
Step 5
Isolate $dy/dx$.
Step 5.1
Subtract $3y$ from both sides.
$$3xy + 2y(dy/dx) = 4 + dy/dx - 3y$$
Step 5.2
Subtract $dy/dx$ from both sides.
$$3xy + 2y(dy/dx) - dy/dx = 4 - 3y$$
Step 5.3
Factor out $dy/dx$.
$$dy/dx(2y - 1) = 4 - 3y - 3xy$$
Step 5.4
Divide both sides by $(2y - 1)$ to solve for $dy/dx$.
$$dy/dx = \frac{4 - 3y - 3xy}{2y - 1}$$
Knowledge Notes:
Sum Rule: The derivative of a sum of functions is the sum of the derivatives.
Product Rule: The derivative of a 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 $nx^{n-1}$.
Differentiating with respect to $x$: When differentiating terms involving $y$, we treat $y$ as a function of $x$ and use the notation $dy/dx$ for its derivative.
Isolating $dy/dx$: To solve for $dy/dx$, we rearrange the equation so that all terms including $dy/dx$ are on one side and then factor and simplify.
