Find dy/dx x^2y-4xy^2=1
The question provided is asking for the derivative of y with respect to x, often denoted as dy/dx, where y is implicitly defined by the equation x^2y - 4xy^2 = 1. This is a problem in calculus involving implicit differentiation, a technique used when a function is not given explicitly as y=f(x), but rather both x and y are mixed together in one equation as shown. The goal is to differentiate both sides of the equation with respect to x, which will involve applying the product rule and the chain rule to differentiate terms containing y. After differentiating, you would typically solve for dy/dx to find the explicit expression for the derivative.
$x^{2} y - 4 x y^{2} = 1$
Answer
Solution:
Step 1:
Take the derivative of both sides of the equation with respect to $x$:
$$\frac{d}{dx}(x^2y - 4xy^2) = \frac{d}{dx}(1)$$
Step 2:
Differentiate the left-hand side of the equation.
Step 2.1:
Apply the Sum Rule in differentiation to the terms $x^2y$ and $-4xy^2$:
$$\frac{d}{dx}(x^2y) + \frac{d}{dx}(-4xy^2)$$
Step 2.2:
Find the derivative of $x^2y$ with respect to $x$.
Step 2.2.1:
Use the Product Rule, which gives the derivative of a product of two functions $f(x)g(x)$ as $f(x)g'(x) + g(x)f'(x)$, where $f(x) = x^2$ and $g(x) = y$:
$$x^2\frac{d}{dx}(y) + y\frac{d}{dx}(x^2) + \frac{d}{dx}(-4xy^2)$$
Step 2.2.2:
Express $\frac{d}{dx}(y)$ as $\frac{dy}{dx}$:
$$x^2\frac{dy}{dx} + y\frac{d}{dx}(x^2) + \frac{d}{dx}(-4xy^2)$$
Step 2.2.3:
Apply the Power Rule, which states that the derivative of $x^n$ is $nx^{n-1}$, where $n = 2$:
$$x^2\frac{dy}{dx} + y(2x) + \frac{d}{dx}(-4xy^2)$$
Step 2.2.4:
Rewrite $2x$ next to $y$:
$$x^2\frac{dy}{dx} + 2xy + \frac{d}{dx}(-4xy^2)$$
Step 2.3:
Find the derivative of $-4xy^2$ with respect to $x$.
Step 2.3.1:
Since $-4$ is a constant, factor it out of the derivative:
$$x^2\frac{dy}{dx} + 2xy - 4\frac{d}{dx}(xy^2)$$
Step 2.3.2:
Apply the Product Rule to $xy^2$, where $f(x) = x$ and $g(x) = y^2$:
$$x^2\frac{dy}{dx} + 2xy - 4\left(x\frac{d}{dx}(y^2) + y^2\frac{d}{dx}(x)\right)$$
Step 2.3.3:
Use the Chain Rule for the derivative of $y^2$, which states that the derivative of $f(g(x))$ is $f'(g(x))g'(x)$, where $f(u) = u^2$ and $g(x) = y$.
Step 2.3.3.1:
Set $u = y$ to apply the Chain Rule:
$$x^2\frac{dy}{dx} + 2xy - 4\left(x\left(\frac{d}{du}(u^2)\frac{d}{dx}(y)\right) + y^2\frac{d}{dx}(x)\right)$$
Step 2.3.3.2:
Differentiate $u^2$ using the Power Rule:
$$x^2\frac{dy}{dx} + 2xy - 4\left(x\left(2u\frac{dy}{dx}\right) + y^2\frac{d}{dx}(x)\right)$$
Step 2.3.3.3:
Replace $u$ back with $y$:
$$x^2\frac{dy}{dx} + 2xy - 4\left(x(2y\frac{dy}{dx}) + y^2\frac{d}{dx}(x)\right)$$
Step 2.3.4:
Express $\frac{d}{dx}(y)$ as $\frac{dy}{dx}$:
$$x^2\frac{dy}{dx} + 2xy - 4\left(x(2y\frac{dy}{dx}) + y^2\right)$$
Step 2.3.5:
Apply the Power Rule to $x$, where $n = 1$:
$$x^2\frac{dy}{dx} + 2xy - 4\left(2xy\frac{dy}{dx} + y^2\right)$$
Step 2.3.6:
Simplify the expression:
$$x^2\frac{dy}{dx} + 2xy - 8xy\frac{dy}{dx} - 4y^2$$
Step 2.3.7:
Multiply $y^2$ by $1$:
$$x^2\frac{dy}{dx} + 2xy - 8xy\frac{dy}{dx} - 4y^2$$
Step 2.4:
Combine all terms:
$$x^2\frac{dy}{dx} + 2xy - 8xy\frac{dy}{dx} - 4y^2$$
Step 3:
The derivative of a constant is zero:
$$0$$
Step 4:
Set the left-hand side equal to the right-hand side:
$$x^2\frac{dy}{dx} + 2xy - 8xy\frac{dy}{dx} - 4y^2 = 0$$
Step 5:
Solve for $\frac{dy}{dx}$.
Step 5.1:
Isolate terms involving $\frac{dy}{dx}$.
Step 5.1.1:
Add $4y^2$ to both sides:
$$x^2\frac{dy}{dx} + 2xy - 8xy\frac{dy}{dx} = 4y^2$$
Step 5.1.2:
Subtract $2xy$ from both sides:
$$x^2\frac{dy}{dx} - 8xy\frac{dy}{dx} = 4y^2 - 2xy$$
Step 5.2:
Factor out $\frac{dy}{dx}$ from the left side.
Step 5.2.1:
Factor $\frac{dy}{dx}$ out of $x^2\frac{dy}{dx}$:
$$\frac{dy}{dx}(x^2 - 8xy) = 4y^2 - 2xy$$
Step 5.3:
Divide both sides by $(x^2 - 8xy)$ to solve for $\frac{dy}{dx}$:
$$\frac{dy}{dx} = \frac{4y^2 - 2xy}{x^2 - 8xy}$$
Knowledge Notes:
Sum Rule: The derivative of a sum of functions is the sum of their derivatives.
Product Rule: The derivative of a product of two functions is given by $f(x)g'(x) + g(x)f'(x)$.
Power Rule: The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
Chain Rule: The derivative of a composite function $f(g(x))$ is $f'(g(x))g'(x)$.
Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
Differentiation of Constants: The derivative of a constant is zero.
In this problem, we applied these rules to differentiate an implicit function given by $x^2y - 4xy^2 = 1$ with respect to $x$. The goal was to find the derivative $\frac{dy}{dx}$, which represents the slope of the tangent line to the curve at any point $(x, y)$. The process involved using the Product Rule and Chain Rule for terms involving both $x$ and $y$, and the Power Rule for terms involving only $x$ or $y$. After differentiating, we rearranged the terms to isolate $\frac{dy}{dx}$ and solve for it explicitly.
