I want questions on implicit and the solution as well

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ScanSolve AI Tutor · Accepted Answer

Okay, here's a question on implicit differentiation and its solution.

Problem: Find $\frac{dy}{dx}$ for the equation $x^3 + y^3 = 6xy$ .

Differentiate both sides with respect to $x$ .
Remember to apply the chain rule for terms involving $y$ and the product rule for $6xy$ .
$\frac{d}{dx}(x^3 + y^3) = \frac{d}{dx}(6xy)$ $\frac{d}{dx}(x^3) + \frac{d}{dx}(y^3) = 6\frac{d}{dx}(xy)$ $3x^2 + 3y^2 \frac{dy}{dx} = 6\left(1 \cdot y + x \cdot \frac{dy}{dx}\right)$ $3x^2 + 3y^2 \frac{dy}{dx} = 6y + 6x \frac{dy}{dx}$
Gather terms containing $\frac{dy}{dx}$ on one side.
Move all terms with $\frac{dy}{dx}$ to the left side and all other terms to the right side.
$3y^2 \frac{dy}{dx} - 6x \frac{dy}{dx} = 6y - 3x^2$
Factor out $\frac{dy}{dx}$ .
$\frac{dy}{dx}(3y^2 - 6x) = 6y - 3x^2$
Solve for $\frac{dy}{dx}$ .
Divide both sides by $(3y^2 - 6x)$ .
$\frac{dy}{dx} = \frac{6y - 3x^2}{3y^2 - 6x}$
We can simplify the expression by dividing the numerator and denominator by 3.
$\frac{dy}{dx} = \frac{3(2y - x^2)}{3(y^2 - 2x)}$ $\frac{dy}{dx} = \frac{2y - x^2}{y^2 - 2x}$

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