Find dy/dx 6x^2+y^2=5

The problem is asking for the derivative of y with respect to x, often denoted as dy/dx. Specifically, it wants you to find this derivative for the implicitly given equation 6x^2 + y^2 = 5. This problem likely requires the use of implicit differentiation because y is not isolated on one side of the equation. Implicit differentiation involves taking the derivative of both sides of the equation with respect to x while treating y as a function of x (y(x)), and then solving for dy/dx.

$6 x^{2} + y^{2} = 5$

Answer

Expert–verified

Solution:

Step:1

Apply differentiation to both sides of the given equation with respect to $x$ : $\frac{d}{d x} (6 x^{2} + y^{2}) = \frac{d}{d x} (5)$ .

Step:2

Differentiate the left-hand side term by term.

Step:2.1

Use the Sum Rule to separate the derivatives: $\frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (y^{2})$ .

Step:2.2

Find the derivative of $6 x^{2}$ with respect to $x$ .

Step:2.2.1

Since $6$ is a constant, it can be factored out: $6 \frac{d}{d x} (x^{2})$ .

Step:2.2.2

Apply the Power Rule, which gives the derivative of $x^{n}$ as $n x^{n - 1}$ for $n = 2$ : $6 (2 x)$ .

Step:2.2.3

Simplify the expression: $12 x$ .

Step:2.3

Now, differentiate $y^{2}$ with respect to $x$ .

Step:2.3.1

Use the Chain Rule, where the derivative of $f (g (x))$ is $f^{'} (g (x)) g^{'} (x)$ , with $f (x) = x^{2}$ and $g (x) = y$ .

Step:2.3.1.1

Let $u = y$ : $12 x + \frac{d}{d u} (u^{2}) \frac{d x}{d y}$ .

Step:2.3.1.2

Differentiate $u^{2}$ using the Power Rule: $12 x + 2 u \frac{d x}{d y}$ .

Step:2.3.1.3

Substitute $u$ back with $y$ : $12 x + 2 y \frac{d x}{d y}$ .

Step:2.3.2

Express $\frac{d x}{d y}$ as $\frac{d y}{d x}$ : $12 x + 2 y y^{'}$ .

Step:2.4

Combine the terms: $2 y y^{'} + 12 x$ .

Step:3

Differentiate the constant $5$ with respect to $x$ : $0$ .

Step:4

Combine the differentiated left-hand side with the right-hand side: $2 y y^{'} + 12 x = 0$ .

Step:5

Isolate $y^{'}$ (which is $\frac{d y}{d x}$ ).

Step:5.1

Subtract $12 x$ from both sides: $2 y y^{'} = - 12 x$ .

Step:5.2

Divide by $2 y$ to solve for $y^{'}$ .

Step:5.2.1

Divide each term by $2 y$ : $\frac{2 y y^{'}}{2 y} = \frac{- 12 x}{2 y}$ .

Step:5.2.2

Simplify both sides.

Step:5.2.2.1

Reduce the fraction on the left-hand side.

Step:5.2.2.1.1

Cancel out the common factor of $2$ : $\frac{y y^{'}}{y} = \frac{- 12 x}{2 y}$ .

Step:5.2.2.1.2

Simplify the expression: $y^{'} = \frac{- 12 x}{2 y}$ .

Step:5.2.2.2

Reduce the fraction on the right-hand side.

Step:5.2.2.2.1

Cancel out the common factor of $y$ : $y^{'} = \frac{- 12 x}{2 y}$ .

Step:5.2.2.2.2

Divide $- 12 x$ by $2 y$ : $y^{'} = \frac{- 6 x}{y}$ .

Step:5.2.3

Simplify the right-hand side further if necessary.

Step:6

Replace $y^{'}$ with $\frac{d y}{d x}$ : $\frac{d y}{d x} = - \frac{6 x}{y}$ .

Knowledge Notes:

To solve the given problem, we used several calculus rules:

Sum Rule: The derivative of a sum is the sum of the derivatives.
Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
Power Rule: The derivative of $x^{n}$ with respect to $x$ is $n x^{n - 1}$ .
Chain Rule: The derivative of a composite function $f (g (x))$ is $f^{'} (g (x)) g^{'} (x)$ .
Differentiation of a Constant: The derivative of a constant is zero.

By applying these rules systematically, we differentiated each term of the given equation with respect to $x$ , isolated the derivative term $\frac{d y}{d x}$ , and solved for it. The process involved algebraic manipulation, such as factoring and canceling common terms, to simplify the expression and obtain the final result.