Find dy/dx x^2+4y^2=9
The problem provided is asking for the derivative of y with respect to x, denoted as dy/dx, for the given equation x^2 + 4y^2 = 9. This is an implicit differentiation problem because y is not isolated on one side of the equation. Implicit differentiation is a technique used to find the derivative of a function that is not expressed as one variable directly in terms of another, which in this case means differentiating both sides of the equation with respect to x, treating y as a function of x (y(x)), and then finding dy/dx from that differentiation.
$x^{2} + 4 y^{2} = 9$
Answer
Solution:
Step 1:
Apply differentiation to both sides of the equation with respect to $x$.
$$\frac{d}{dx}(x^2 + 4y^2) = \frac{d}{dx}(9)$$
Step 2:
Differentiate the left-hand side term by term.
Step 2.1:
Differentiate each term.
Step 2.1.1:
Utilize the Sum Rule in differentiation: the derivative of a sum is the sum of the derivatives.
$$\frac{d}{dx}(x^2) + \frac{d}{dx}(4y^2)$$
Step 2.1.2:
Apply the Power Rule for differentiation: $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n$ is a constant.
$$2x + \frac{d}{dx}(4y^2)$$
Step 2.2:
Determine the derivative of $4y^2$ with respect to $x$.
Step 2.2.1:
Recognize that $4$ is a constant and can be factored out of the derivative.
$$2x + 4\frac{d}{dx}(y^2)$$
Step 2.2.2:
Employ the Chain Rule for differentiation: $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$.
Step 2.2.2.1:
Let $u = y$ to prepare for the Chain Rule.
$$2x + 4\left(\frac{d}{du}(u^2)\frac{d}{dx}(y)\right)$$
Step 2.2.2.2:
Apply the Power Rule to $u^2$.
$$2x + 4(2u\frac{d}{dx}(y))$$
Step 2.2.2.3:
Substitute $y$ back in for $u$.
$$2x + 4(2y\frac{d}{dx}(y))$$
Step 2.2.3:
Express $\frac{d}{dx}(y)$ as $dy/dx$.
$$2x + 8y(dy/dx)$$
Step 2.3:
Rearrange the terms for clarity.
$$8y(dy/dx) + 2x$$
Step 3:
Differentiate the right-hand side, noting that the derivative of a constant is zero.
$$0$$
Step 4:
Combine the differentiated left and right sides into one equation.
$$8y(dy/dx) + 2x = 0$$
Step 5:
Isolate $dy/dx$ to solve for the derivative.
Step 5.1:
Subtract $2x$ from both sides.
$$8y(dy/dx) = -2x$$
Step 5.2:
Divide both sides by $8y$ to isolate $dy/dx$.
Step 5.2.1:
Divide both terms by $8y$.
$$\frac{8y(dy/dx)}{8y} = \frac{-2x}{8y}$$
Step 5.2.2:
Simplify the left side by canceling out common factors.
Step 5.2.2.1:
Cancel the $8y$ terms.
$$\frac{y(dy/dx)}{y} = \frac{-2x}{8y}$$
Step 5.2.2.2:
Reduce the fraction on the left side.
$$(dy/dx) = \frac{-2x}{8y}$$
Step 5.2.3:
Simplify the right side by reducing the fraction.
Step 5.2.3.1:
Factor out and cancel common factors.
$$(dy/dx) = \frac{-x}{4y}$$
Step 5.2.3.2:
Place the negative sign in front of the fraction.
$$(dy/dx) = -\frac{x}{4y}$$
Step 6:
Finalize the expression for $\frac{dy}{dx}$.
$$\frac{dy}{dx} = -\frac{x}{4y}$$
Knowledge Notes:
The problem involves finding the derivative of an implicitly defined function. Implicit differentiation is used when a function is not given explicitly as $y=f(x)$. The steps involve:
Differentiation of both sides of the equation with respect to $x$.
Application of differentiation rules such as the Sum Rule, Power Rule, and Chain Rule.
Rearrangement and simplification of the equation to solve for $\frac{dy}{dx}$.
The Sum Rule states that the derivative of a sum is the sum of the derivatives. The Power Rule states that the derivative of $x^n$ is $nx^{n-1}$. The Chain Rule is used to differentiate composite functions and states that the derivative of $f(g(x))$ is $f'(g(x))g'(x)$. When applying these rules, it's important to treat $y$ as a function of $x$ and use the Chain Rule to differentiate terms involving $y$.
