3.8 Implicit Differentiation; Questions 300-309

[Eigenvalue]

For the following exercises, use implicit differentiation to find [tex]\frac{dy}{dx}[/tex]

300. x²-y²=4
301. 6x²+3y²=12
302. x²y=y-7
303. 3x³-9xy²=5x³
304. xy-cos(xy)=1
305. y[tex]\sqrt{x+4}[/tex]=xy+8
306. -xy-2=[tex]\frac{x}{7}[/tex]
307. ysin(xy)=y²-2
308. (xy)²+3x=y²
309. x³y+xy³=-8

[Eigenvalue]

300. x²-y²=4
Differentiate both sides: (x²-y²)'=(4)'
2x-2y[tex]\frac{dy}{dx}[/tex]=0

2x=2y[tex]\frac{dy}{dx}[/tex]

[tex]\frac{dy}{dx}[/tex]=[tex]\frac{x}{y}[/tex]

301. 6x²+3y²=12
Differentiate both sides: (6x²+3y²)'=(12)'

12x+6y[tex]\frac{dy}{dx}[/tex]=0

6y[tex]\frac{dy}{dx}[/tex]=-12x

[tex]\frac{dy}{dx}[/tex]=[tex]\frac{-2x}{y}[/tex]

302. x²y=y-7
Differentiate both sides: (x²y)'=(y-7)'

x²[tex]\frac{dy}{dx}[/tex]+2xy=[tex]\frac{dy}{dx}[/tex]

(x²-1)[tex]\frac{dy}{dx}[/tex]=-2xy

[tex]\frac{dy}{dx}[/tex]=[tex]\frac{-2xy}{x²-1}[/tex]


303. 3x³-9xy²=5x³
Simplify: -9xy²=2x³
Differentiate both sides: (-9xy²)'=(2x³)'
-18xy[tex]\frac{dy}{dx}[/tex]-9y²=6x²

[tex]\frac{dy}{dx}[/tex]=[tex]\frac{6x²+9y²}{-18xy}[/tex]=[tex]\frac{-2x²-3y²}{6xy}[/tex]