Question

differentiate implicitly to find dy/dx. then find the slope of the curve at the given point.

$x^{2}y^{2}=49$, $(1,7)$

$\frac{dy}{dx}=\square$

Explanation:

Step1: Differentiate both sides with respect to \(x\)

Use the product rule \((uv)^\prime = u^\prime v+uv^\prime\) where \(u = x^{2}\) and \(v = y^{2}\).
Differentiating \(x^{2}y^{2}\) gives \(2xy^{2}+2x^{2}y\frac{dy}{dx}\), and differentiating \(49\) gives \(0\). So, \(2xy^{2}+2x^{2}y\frac{dy}{dx}=0\).

Step2: Solve for \(\frac{dy}{dx}\)

First, factor out \(2xy\) from the left - hand side: \(2xy(y + x\frac{dy}{dx})=0\).
Since \(x = 1\) and \(y = 7\) (non - zero values at the given point \((1,7)\)), we can solve \(2xy^{2}+2x^{2}y\frac{dy}{dx}=0\) for \(\frac{dy}{dx}\) directly.
Subtract \(2xy^{2}\) from both sides: \(2x^{2}y\frac{dy}{dx}=- 2xy^{2}\).
Then divide both sides by \(2x^{2}y\) (\(x
eq0,y
eq0\)): \(\frac{dy}{dx}=-\frac{y}{x}\).

Step3: Find the slope at the point \((1,7)\)

Substitute \(x = 1\) and \(y = 7\) into \(\frac{dy}{dx}=-\frac{y}{x}\).
\(\frac{dy}{dx}\big|_{(1,7)}=-\frac{7}{1}=-7\).

Answer:

\(\frac{dy}{dx}=-\frac{y}{x}\), and the slope at the point \((1,7)\) is \(-7\).