$$
x^{3} y+y^{2}-x^{2}=5
$$

Find the value of $\frac{d y}{d x}$ at the point $(2,1)$.

Question

$$
x^{3} y+y^{2}-x^{2}=5
$$

Find the value of $\frac{d y}{d x}$ at the point $(2,1)$.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Differentiate the given equation implicitly with respect to $x$: $x^{3} y+y^{2}-x^{2}=5$. ‖ ‖ step 2 ⋮ Apply the product rule to $x^3 y$ and the power rule to $y^2$ and $x^2$. ‖ ‖ step 3 ⋮ Set the derivative equal to zero and solve for $\frac{dy}{dx}$. ‖ ‖ step 4 ⋮ Substitute $x=2$ and $y=1$ into the derivative to find the value of $\frac{dy}{dx}$ at the point $(2,1)$. ‖ ∻Answer∻ ⚹ $\frac{dy}{dx} = -\frac{1}{3}$ at $(2,1)$ ⚹ ∻Key Concept∻ ⚹Implicit Differentiation⚹ ∻Explanation∻ ⚹To find $\frac{dy}{dx}$ for an equation involving both $x$ and $y$, we differentiate both sides of the equation with respect to $x$, treating $y$ as an implicit function of $x$.⚹◊What is the derivative of $x^{3} y+y^{2}-x^{2}=5$ with respect to $x$?⍭ Generate me a similar question◊

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