Question

use implicit differentiation of the equation below to determine the slope of the g

xy^{4}=-49;x = -\frac{1}{49},y = -7

use implicit differentiation to determine \frac{dy}{dx}

\frac{dy}{dx}=\square

Explanation:

Step1: Differentiate both sides

Differentiate \(xy^{4}=-49\) with respect to \(x\) using the product rule \((uv)^\prime = u^\prime v+uv^\prime\) (where \(u = x\), \(v = y^{4}\)).
The derivative of \(x\) with respect to \(x\) is \(1\), and for \(y^{4}\) with respect to \(x\), we use the chain rule \(\frac{d}{dx}(y^{4})=4y^{3}\frac{dy}{dx}\).
So, \(\frac{d}{dx}(xy^{4})=\frac{d}{dx}(-49)\).
\(y^{4}+x\cdot4y^{3}\frac{dy}{dx}=0\).

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

Rearrange the equation \(y^{4}+4xy^{3}\frac{dy}{dx}=0\) for \(\frac{dy}{dx}\).
First, move \(y^{4}\) to the other side: \(4xy^{3}\frac{dy}{dx}=-y^{4}\).
Then, divide both sides by \(4xy^{3}\) (assuming \(x
eq0\) and \(y
eq0\)): \(\frac{dy}{dx}=-\frac{y}{4x}\).

Step3: Substitute \(x =-\frac{1}{49}\) and \(y = - 7\)

Substitute \(x =-\frac{1}{49}\) and \(y=-7\) into \(\frac{dy}{dx}=-\frac{y}{4x}\).
\(\frac{dy}{dx}=-\frac{-7}{4\times(-\frac{1}{49})}\).
\(\frac{dy}{dx}=-\frac{7}{\frac{4}{49}}\).
Using the rule \(\frac{a}{b/a}=a\times\frac{a}{b}\), we have \(\frac{dy}{dx}=-\frac{7\times49}{4}=-\frac{343}{4}\).

Answer:

\(-\frac{343}{4}\)