Find dy/dx y^2-4xy+x^2=10
The problem you've presented is asking for the differentiation of an implicitly defined function with respect to \( x \). Implicit differentiation is a technique used when a function \( y \) is not given in the standard form of \( y = f(x) \) but rather is mixed with \( x \) in an equation. Here, you have an equation \( y^2 - 4xy + x^2 = 10 \) involving both \( y \) and \( x \), and you need to differentiate each term with respect to \( x \), while keeping in mind that \( y \) is a function of \( x \). This will require the use of the chain rule for terms involving \( y \), as \( dy/dx \) will need to be accounted for in those terms.
$y^{2} - 4 x y + x^{2} = 10$
Answer
Solution:
Step 1:
Take the derivative of both sides of the equation $y^2 - 4xy + x^2 = 10$ with respect to $x$.
Step 2:
Differentiate the left-hand side term by term.
Step 2.1:
Apply the Sum Rule to find the derivative of the sum of functions: $\frac{d}{dx}(y^2) + \frac{d}{dx}(-4xy) + \frac{d}{dx}(x^2)$.
Step 2.2:
Compute $\frac{d}{dx}(y^2)$.
Step 2.2.1:
Use the Chain Rule, where the derivative of a composite function $f(g(x))$ is $f'(g(x))g'(x)$. Here, $f(x) = x^2$ and $g(x) = y$.
Step 2.2.1.1:
Introduce a substitution $u = y$ and differentiate: $\frac{d}{du}(u^2) \frac{dy}{dx} + \frac{d}{dx}(-4xy) + \frac{d}{dx}(x^2)$.
Step 2.2.1.2:
Apply the Power Rule, which states that $\frac{d}{du}(u^n) = nu^{n-1}$ for $n = 2$: $2u \frac{dy}{dx} + \frac{d}{dx}(-4xy) + \frac{d}{dx}(x^2)$.
Step 2.2.1.3:
Substitute $y$ back in for $u$: $2y \frac{dy}{dx} + \frac{d}{dx}(-4xy) + \frac{d}{dx}(x^2)$.
Step 2.2.2:
Recognize that $\frac{d}{dx}(y)$ is $\frac{dy}{dx}$: $2y \frac{dy}{dx} + \frac{d}{dx}(-4xy) + \frac{d}{dx}(x^2)$.
Step 2.3:
Compute $\frac{d}{dx}(-4xy)$.
Step 2.3.1:
Extract the constant $-4$ from the derivative: $2y \frac{dy}{dx} - 4 \frac{d}{dx}(xy) + \frac{d}{dx}(x^2)$.
Step 2.3.2:
Differentiate $xy$ using the Product Rule, which gives the derivative of the product of two functions $f(x)g(x)$ as $f(x)g'(x) + g(x)f'(x)$: $2y \frac{dy}{dx} - 4(x \frac{dy}{dx} + y) + \frac{d}{dx}(x^2)$.
Step 2.3.3:
Replace $\frac{d}{dx}(y)$ with $\frac{dy}{dx}$: $2y \frac{dy}{dx} - 4(x \frac{dy}{dx} + y) + \frac{d}{dx}(x^2)$.
Step 2.3.4:
Differentiate $x$ using the Power Rule: $2y \frac{dy}{dx} - 4(x \frac{dy}{dx} + y \cdot 1) + \frac{d}{dx}(x^2)$.
Step 2.3.5:
Simplify the multiplication by 1: $2y \frac{dy}{dx} - 4(xy + y) + \frac{d}{dx}(x^2)$.
Step 2.4:
Apply the Power Rule to $\frac{d}{dx}(x^2)$: $2y \frac{dy}{dx} - 4(xy + y) + 2x$.
Step 2.5:
Combine and simplify the terms: $2y \frac{dy}{dx} - 4xy - 4y + 2x$.
Step 3:
Differentiate the constant 10 with respect to $x$: $0$.
Step 4:
Combine the derivatives from the left and right sides to form the equation: $2y \frac{dy}{dx} - 4xy + 2x - 4y = 0$.
Step 5:
Isolate $\frac{dy}{dx}$.
Step 5.1:
Transpose terms not containing $\frac{dy}{dx}$ to the other side: $2y \frac{dy}{dx} - 4xy = -2x + 4y$.
Step 5.2:
Factor out $2y$ from the left side: $2y(\frac{dy}{dx} - 2x) = -2x + 4y$.
Step 5.3:
Divide both sides by $2y - 2x$ to solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{-2x + 4y}{2y - 2x}$.
Step 6:
Substitute $\frac{dy}{dx}$ back into the equation: $\frac{dy}{dx} = -\frac{x - 2y}{y - 2x}$.
Knowledge Notes:
Sum Rule: The derivative of a sum of functions is the sum of the derivatives.
Chain Rule: The derivative of a composite function $f(g(x))$ is $f'(g(x))g'(x)$.
Power Rule: The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
Product Rule: The derivative of the product of two functions $f(x)g(x)$ is $f(x)g'(x) + g(x)f'(x)$.
Implicit Differentiation: When a function is not given explicitly as $y=f(x)$, but rather in a form involving both $x$ and $y$, we differentiate both sides of the equation with respect to $x$ and solve for $\frac{dy}{dx}$.
Differentiating Constants: The derivative of a constant is zero.
Transposition: Moving terms from one side of an equation to the other to isolate a variable or expression.
Factoring: Expressing an expression as a product of its factors to simplify the equation or solve for a variable.
