Question

suppose that $x^{4}+y^{4}=82$.
(1) use the method of implicit differentiation to find $\frac{dy}{dx}$.
$\frac{dy}{dx}=$
(2) find the equation of the tangent line at the point $(x,y)=(3,-1)$.
the equation is $y=$
question help: video

Explanation:

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

Differentiate \(x^{4}+y^{4}=82\).
Using the power rule \((u^{n})^\prime = nu^{n - 1}u^\prime\), we have:
\(\frac{d}{dx}(x^{4})+\frac{d}{dx}(y^{4})=\frac{d}{dx}(82)\)
\(4x^{3}+4y^{3}\frac{dy}{dx}=0\)

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

Subtract \(4x^{3}\) from both sides: \(4y^{3}\frac{dy}{dx}=- 4x^{3}\)
Divide both sides by \(4y^{3}\): \(\frac{dy}{dx}=-\frac{x^{3}}{y^{3}}\)

Step3: Find the slope of the tangent line at \((3,-1)\)

Substitute \(x = 3\) and \(y=-1\) into \(\frac{dy}{dx}\):
\(m=\frac{dy}{dx}\big|_{x = 3,y=-1}=-\frac{3^{3}}{(-1)^{3}}=27\)

Step4: Use the point - slope form \(y - y_{1}=m(x - x_{1})\)

Here \(x_{1}=3,y_{1}=-1,m = 27\)
\(y+1=27(x - 3)\)
Expand: \(y+1=27x-81\)
Solve for \(y\): \(y = 27x-82\)

Answer:

1. \(\frac{dy}{dx}=-\frac{x^{3}}{y^{3}}\)
2. \(y = 27x-82\)