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Find dy/dx for 3y^5 = 2x^3 + y using Implicit Differentiation
Mathematics
Grade 11 (Senior High School)
Question Content
Find $\frac{dy}{dx}$ for $3y^5 = 2x^3 + y$
Correct Answer
$\\frac{dy}{dx} = \\frac{6x^2}{15y^4 - 1}$
Detailed Solution Steps
1
Step 1: Differentiate both sides of the equation with respect to $x$. Use implicit differentiation: for the left side, apply the chain rule to $3y^5$ to get $15y^4\\frac{dy}{dx}$; for the right side, differentiate $2x^3$ to get $6x^2$ and $y$ to get $\\frac{dy}{dx}$. This gives $15y^4\\frac{dy}{dx} = 6x^2 + \\frac{dy}{dx}$.
2
Step 2: Collect all terms with $\\frac{dy}{dx}$ on one side. Subtract $\\frac{dy}{dx}$ from both sides: $15y^4\\frac{dy}{dx} - \\frac{dy}{dx} = 6x^2$.
3
Step 3: Factor out $\\frac{dy}{dx}$ from the left side: $\\frac{dy}{dx}(15y^4 - 1) = 6x^2$.
4
Step 4: Solve for $\\frac{dy}{dx}$ by dividing both sides by $15y^4 - 1$: $\\frac{dy}{dx} = \\frac{6x^2}{15y^4 - 1}$.
Knowledge Points Involved
1
Implicit Differentiation
A method to find the derivative of a function that is not explicitly solved for $y$. It involves differentiating both sides of the equation with respect to $x$, treating $y$ as a function of $x$ and applying the chain rule to terms involving $y$.
2
Chain Rule
A differentiation rule that states if $y = f(g(x))$, then $\\frac{dy}{dx} = f'(g(x)) \\cdot g'(x)$. When differentiating terms with $y$ (like $y^5$), we treat $y$ as a function of $x$, so $\\frac{d}{dx}(y^5) = 5y^4\\frac{dy}{dx}$.
3
Power Rule for Differentiation
A basic differentiation rule that states $\\frac{d}{dx}(x^n) = nx^{n-1}$ where $n$ is a real number. Used here to differentiate $2x^3$ to get $6x^2$.
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