suppose that $x^{4}+y^{4}=82$.\n(1) use the method of implicit differentiation to find $\frac{dy}{dx}$.\n$\fr…

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

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

Answer

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)