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Calculate dy/dx for x - 3y^2 = 6 using Implicit Differentiation
Mathematics
Grade 11 (Senior High School)
Question Content
Find $\frac{dy}{dx}$ for $x - 3y^2 = 6$
Correct Answer
$\\frac{dy}{dx} = \\frac{1}{6y}$
Detailed Solution Steps
1
Step 1: Differentiate both sides with respect to $x$. Differentiate $x$ to get $1$, $-3y^2$ to get $-6y\\frac{dy}{dx}$, and $6$ to get $0$. This gives $1 - 6y\\frac{dy}{dx} = 0$.
2
Step 2: Rearrange to isolate the term with $\\frac{dy}{dx}$: $-6y\\frac{dy}{dx} = -1$.
3
Step 3: Divide both sides by $-6y$ to solve for $\\frac{dy}{dx}$: $\\frac{dy}{dx} = \\frac{1}{6y}$.
Knowledge Points Involved
1
Implicit Differentiation
Used to differentiate equations where $y$ is not isolated, applying the chain rule to terms with $y$.
2
Chain Rule for Power Functions
When differentiating $-3y^2$ with respect to $x$, the derivative is $-6y\\frac{dy}{dx}$, as we treat $y$ as a function of $x$ and apply the power rule combined with the chain rule.
3
Derivative of a Constant
The derivative of a constant (like 6) with respect to $x$ is 0, since constants do not change with $x$.
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