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

Expert–verified

Solution:

Step 1

Take the derivative of both sides of the equation with respect to $x$ : $\frac{d}{d x} (x^{2} + x y + 2 y^{2}) = \frac{d}{d x} (16)$

Step 2

Differentiate the left-hand side term by term.

Step 2.1

Apply the Sum Rule to separate the derivatives: $\frac{d}{d x} (x^{2}) + \frac{d}{d x} (x y) + \frac{d}{d x} (2 y^{2})$

Step 2.1.1

Use the Power Rule, where the derivative of $x^{n}$ is $n x^{n - 1}$ , to find $\frac{d}{d x} (x^{2}) = 2 x$ .

Step 2.1.2

The derivative of $x^{2}$ is $2 x$ : $2 x + \frac{d}{d x} (x y) + \frac{d}{d x} (2 y^{2})$

Step 2.2

Differentiate $x y$ using the Product Rule.

Step 2.2.1

The Product Rule states that $\frac{d}{d x} (f (x) g (x)) = f^{'} (x) g (x) + f (x) g^{'} (x)$ , so for $f (x) = x$ and $g (x) = y$ , we get: $2 x + x \frac{d}{d x} (y) + y \frac{d}{d x} (x) + \frac{d}{d x} (2 y^{2})$

Step 2.2.2

Replace $\frac{d}{d x} (y)$ with $\frac{d y}{d x}$ : $2 x + x \frac{d y}{d x} + y + \frac{d}{d x} (2 y^{2})$

Step 2.2.3

Apply the Power Rule to $\frac{d}{d x} (x)$ , which is $1$ : $2 x + x \frac{d y}{d x} + y + \frac{d}{d x} (2 y^{2})$

Step 2.3

Differentiate $2 y^{2}$ .

Step 2.3.1

As $2$ is a constant, it remains unchanged: $2 x + x \frac{d y}{d x} + y + 2 \frac{d}{d x} (y^{2})$

Step 2.3.2

Apply the Chain Rule to $\frac{d}{d x} (y^{2})$ , where $f (u) = u^{2}$ and $u = y$ : $2 x + x \frac{d y}{d x} + y + 2 (2 y \frac{d y}{d x})$

Step 2.4

Combine all terms: $2 x + x \frac{d y}{d x} + y + 4 y \frac{d y}{d x}$

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: $2 x + x \frac{d y}{d x} + y + 4 y \frac{d y}{d x} = 0$

Step 5

Solve for $\frac{d y}{d x}$ .

Step 5.1

Isolate terms involving $\frac{d y}{d x}$ on one side: $x \frac{d y}{d x} + 4 y \frac{d y}{d x} = - 2 x - y$

Step 5.2

Factor out $\frac{d y}{d x}$ : $\frac{d y}{d x} (x + 4 y) = - 2 x - y$

Step 5.3

Divide both sides by $(x + 4 y)$ to solve for $\frac{d y}{d x}$ : $\frac{d y}{d x} = \frac{- 2 x - y}{x + 4 y}$

Step 6

The final result is: $\frac{d y}{d x} = - \frac{2 x + y}{x + 4 y}$

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 $n x^{n - 1}$ .
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 $\frac{d y}{d x}$ .
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 $\frac{d y}{d x}$ .