Find dy/dx 2x^2-xy+y^2=4

Solution:

Step:1

Take the derivative of both sides of the equation with respect to $x$: $\frac{d}{dx}(2x^2 - xy + y^2) = \frac{d}{dx}(4)$.

Step:2

Differentiate each term on the left side individually.

Step:2.1

Apply the Sum Rule in differentiation: the derivative of a sum is the sum of the derivatives. Thus, $\frac{d}{dx}(2x^2) + \frac{d}{dx}(-xy) + \frac{d}{dx}(y^2)$.

Step:2.2

Find the derivative of $2x^2$ with respect to $x$.

Step:2.2.1

The constant $2$ remains unchanged during differentiation, so we have $2\frac{d}{dx}(x^2)$.

Step:2.2.2

Apply the Power Rule, which gives us $2(2x)$.

Step:2.2.3

Simplify to get $4x$.

Step:2.3

Find the derivative of $-xy$ with respect to $x$.

Step:2.3.1

The constant $-1$ remains unchanged during differentiation, so we have $-\frac{d}{dx}(xy)$.

Step:2.3.2

Apply the Product Rule: $\frac{d}{dx}(x)y + x\frac{d}{dx}(y)$.

Step:2.3.3

Since $\frac{d}{dx}(y)$ is the derivative of $y$ with respect to $x$, denote it as $y'$.

Step:2.3.4

Differentiate $x$ with respect to $x$ to get $1$.

Step:2.3.5

Simplify to $-xy' - y$.

Step:2.4

Find the derivative of $y^2$ with respect to $x$.

Step:2.4.1

Apply the Chain Rule: $\frac{d}{dx}(y^2) = 2y\frac{dy}{dx}$.

Step:2.5

Combine all the derived terms: $4x - xy' - y + 2y\frac{dy}{dx}$.

Step:3

Differentiate the constant $4$ on the right side to get $0$.

Step:4

Combine the derivatives to form the equation: $4x - xy' - y + 2y\frac{dy}{dx} = 0$.

Step:5

Isolate $\frac{dy}{dx}$.

Step:5.1

Move terms that do not contain $\frac{dy}{dx}$ to the other side: $-xy' + 2y\frac{dy}{dx} = -4x + y$.

Step:5.2

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

Step:5.3

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

Knowledge Notes:

1. Sum Rule: The derivative of a sum of functions is the sum of the derivatives of those functions.

2. Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

3. Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

4. Product Rule: If $u(x)$ and $v(x)$ are functions of $x$, then the derivative of their product $u(x)v(x)$ is given by $u'(x)v(x) + u(x)v'(x)$.

5. Chain Rule: If a variable $y$ is a function of $u$ which is a function of $x$, then the derivative of $y$ with respect to $x$ is given by $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

6. Implicit Differentiation: When a function is not given in the form $y = f(x)$, but instead involves $y$ and $x$ in an equation, we differentiate both sides of the equation with respect to $x$ and solve for $\frac{dy}{dx}$.

7. Differentiating Constants: The derivative of a constant is zero.

8. Solving for $\frac{dy}{dx}$: After differentiating, we often need to isolate $\frac{dy}{dx}$ to find the slope of the tangent line at any point $(x, y)$ on the curve defined by the given equation.