Consider the curve $xy-2x=-9$. Evaluate $\frac{dy}{dx}$ at $(1,-7)$

Key concept: Because $y$ is defined implicitly, differentiate both sides with respect to $x$ and then plug in the point $(1,-7)$.

Step by step

Step 1: Differentiate implicitly

Start with

$xy-2x=-9$

Differentiate both sides with respect to $x$:

• For $xy$, use the product rule:
  $\frac{d}{dx}(xy)=x\frac{dy}{dx}+y$
• For $-2x$: $\frac{d}{dx}(-2x)=-2$
• The derivative of $-9$ is $0$.

So,

$x\frac{dy}{dx}+y-2=0$

Step 2: Solve for $\frac{dy}{dx}$

$x\frac{dy}{dx}=2-y$

$\frac{dy}{dx}=\frac{2-y}{x}$

Step 3: Evaluate at $(1,-7)$

$\frac{dy}{dx}\Big|_{(1,-7)}=\frac{2-(-7)}{1}=9$

Final answer

$\boxed{9}$

Pitfall alert

Do not differentiate $xy$ as $x\cdot y'$ only. The product rule is required because both $x$ and $y$ depend on the curve relationship.

Try different conditions

If the point were different, you would use the same derivative formula $\frac{dy}{dx}=\frac{2-y}{x}$ and substitute the new coordinates. If $x=0$, you would need to check the original equation carefully because the formula would not be defined there.

Further reading

implicit differentiation, product rule, slope