Question

Math

Posted 5 months ago

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

Posted 5 months ago

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 $x$.

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