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
Solution:
Step:1
Apply differentiation to both sides of the given equation with respect to $x$: $\frac{d}{dx}(6x^2 + y^2) = \frac{d}{dx}(5)$.
Step:2
Differentiate the left-hand side term by term.
Step:2.1
Use the Sum Rule to separate the derivatives: $\frac{d}{dx}(6x^2) + \frac{d}{dx}(y^2)$.
Step:2.2
Find the derivative of $6x^2$ with respect to $x$.
Step:2.2.1
Since $6$ is a constant, it can be factored out: $6\frac{d}{dx}(x^2)$.
Step:2.2.2
Apply the Power Rule, which gives the derivative of $x^n$ as $nx^{n-1}$ for $n=2$: $6(2x)$.
Step:2.2.3
Simplify the expression: $12x$.
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 $u = y$: $12x + \frac{d}{du}(u^2)\frac{dx}{dy}$.
Step:2.3.1.2
Differentiate $u^2$ using the Power Rule: $12x + 2u\frac{dx}{dy}$.
Step:2.3.1.3
Substitute $u$ back with $y$: $12x + 2y\frac{dx}{dy}$.
Step:2.3.2
Express $\frac{dx}{dy}$ as $\frac{dy}{dx}$: $12x + 2yy'$.
Step:2.4
Combine the terms: $2yy' + 12x$.
Step:3
Differentiate the constant $5$ with respect to $x$: $0$.
Step:4
Combine the differentiated left-hand side with the right-hand side: $2yy' + 12x = 0$.
Step:5
Isolate $y'$ (which is $\frac{dy}{dx}$).
Step:5.1
Subtract $12x$ from both sides: $2yy' = -12x$.
Step:5.2
Divide by $2y$ to solve for $y'$.
Step:5.2.1
Divide each term by $2y$: $\frac{2yy'}{2y} = \frac{-12x}{2y}$.
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 $2$: $\frac{yy'}{y} = \frac{-12x}{2y}$.
Step:5.2.2.1.2
Simplify the expression: $y' = \frac{-12x}{2y}$.
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 $y$: $y' = \frac{-12x}{2y}$.
Step:5.2.2.2.2
Divide $-12x$ by $2y$: $y' = \frac{-6x}{y}$.
Step:5.2.3
Simplify the right-hand side further if necessary.
Step:6
Replace $y'$ with $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{6x}{y}$.
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 $nx^{n-1}$.
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 $\frac{dy}{dx}$, 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.
