Find dy/dx x^2+xy+2y^2=16
This question is asking to find the derivative of y with respect to x, commonly denoted as dy/dx, from the given implicit equation x^2 + xy + 2y^2 = 16. In contrast to explicit functions where y is given in terms of x (e.g., y = f(x)), an implicit function involves an equation where y cannot be easily isolated on one side. The task requires using implicit differentiation, which is a technique where both sides of the equation are differentiated with respect to x, and the chain rule is applied whenever a term involving y is differentiated, since y is itself a function of x.
$x^{2} + x y + 2 y^{2} = 16$
Answer
Solution:
Step 1
Take the derivative of both sides of the equation with respect to $x$: $\frac{d}{dx}(x^2 + xy + 2y^2) = \frac{d}{dx}(16)$
Step 2
Differentiate the left-hand side term by term.
Step 2.1
Apply the Sum Rule to separate the derivatives: $\frac{d}{dx}(x^2) + \frac{d}{dx}(xy) + \frac{d}{dx}(2y^2)$
Step 2.1.1
Use the Power Rule, where the derivative of $x^n$ is $nx^{n-1}$, to find $\frac{d}{dx}(x^2) = 2x$.
Step 2.1.2
The derivative of $x^2$ is $2x$: $2x + \frac{d}{dx}(xy) + \frac{d}{dx}(2y^2)$
Step 2.2
Differentiate $xy$ using the Product Rule.
Step 2.2.1
The Product Rule states that $\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$, so for $f(x) = x$ and $g(x) = y$, we get: $2x + x\frac{d}{dx}(y) + y\frac{d}{dx}(x) + \frac{d}{dx}(2y^2)$
Step 2.2.2
Replace $\frac{d}{dx}(y)$ with $\frac{dy}{dx}$: $2x + x\frac{dy}{dx} + y + \frac{d}{dx}(2y^2)$
Step 2.2.3
Apply the Power Rule to $\frac{d}{dx}(x)$, which is $1$: $2x + x\frac{dy}{dx} + y + \frac{d}{dx}(2y^2)$
Step 2.3
Differentiate $2y^2$.
Step 2.3.1
As $2$ is a constant, it remains unchanged: $2x + x\frac{dy}{dx} + y + 2\frac{d}{dx}(y^2)$
Step 2.3.2
Apply the Chain Rule to $\frac{d}{dx}(y^2)$, where $f(u) = u^2$ and $u = y$: $2x + x\frac{dy}{dx} + y + 2(2y\frac{dy}{dx})$
Step 2.4
Combine all terms: $2x + x\frac{dy}{dx} + y + 4y\frac{dy}{dx}$
Step 3
The derivative of a constant, $16$, is $0$: $0$
Step 4
Set the derivative of the left-hand side equal to the derivative of the right-hand side: $2x + x\frac{dy}{dx} + y + 4y\frac{dy}{dx} = 0$
Step 5
Solve for $\frac{dy}{dx}$.
Step 5.1
Isolate terms involving $\frac{dy}{dx}$ on one side: $x\frac{dy}{dx} + 4y\frac{dy}{dx} = -2x - y$
Step 5.2
Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(x + 4y) = -2x - y$
Step 5.3
Divide both sides by $(x + 4y)$ to solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{-2x - y}{x + 4y}$
Step 6
The final result is: $\frac{dy}{dx} = -\frac{2x + y}{x + 4y}$
Knowledge Notes:
Sum Rule: The derivative of a sum of functions is the sum of the derivatives of each function.
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 $f(x)$ and $g(x)$ is given by $f'(x)g(x) + f(x)g'(x)$.
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.
Implicit Differentiation: When a function is not given explicitly as $y=f(x)$, but instead involves $y$ and $x$ in an equation, we differentiate with respect to $x$ and solve for $\frac{dy}{dx}$.
Derivative of a Constant: The derivative of a constant is zero.
The problem-solving process involves applying these rules of differentiation to find the derivative of the given equation implicitly, and then solving for $\frac{dy}{dx}$.
