Answer to Question #262721 in Calculus for Alunsina

Question #262721

Find y'. Simplify all answers.

a. x²y + xy² - y³ + x² = 2

b. sqrt(2xy) + y²-xy²=1

c. sinx + cosy = sinxcosy

d. y = x^x^x

e. y = (sinx)^2x

f. ln(x²+y²) + 2arctan(y/x) = 1

g. (x+y)³ = (x-y)²


1
Expert's answer
2021-11-22T07:33:30-0500

a.


(x2y+xy2y3+x2)=(2)(x^2y + xy^2- y^3+ x^2 )' = (2)'

2xy+x2y+y2+2xyy3y2y+2x=02xy+x^2y'+y^2+2xyy'-3y^2y'+2x=0

y=2xy+y2+2x3y22xyx2y'=\dfrac{2xy+y^2+2x}{3y^2-2xy-x^2}

b.


(2xy+y2xy2)=(1)(\sqrt{2xy} + y^2-xy^2)'=(1)'

y2x+x2yy+2yyy22xyy=0\sqrt{\dfrac{y}{2x}}+\sqrt{\dfrac{x}{2y}}y'+2yy'-y^2-2xyy'=0

y=y2y2xx2y+2y2xyy'=\dfrac{y^2-\sqrt{\dfrac{y}{2x}}}{\sqrt{\dfrac{x}{2y}}+2y-2xy}

c.


(sinx+cosy)=(sinxcosy)(\sin x + \cos y )'= (\sin x\cos y)'

cosxysiny=cosxcosyysinxsiny\cos x-y'\sin y=\cos x\cos y-y'\sin x \sin y

y=cosxcosycosxsinxsinysinyy'=\dfrac{\cos x\cos y-\cos x}{\sin x \sin y-\sin y}

d.


y=xxxy = x^{x^x}

lny=xxlnx\ln y=x^x\ln x

lnylnx=xx\dfrac{\ln y}{\ln x}=x^x

ln(lnylnx)=xlnx\ln(\dfrac{\ln y}{\ln x})=x\ln x

(ln(lny)ln(lnx))=(xlnx)(\ln(\ln y)-\ln(\ln x))'=(x\ln x)'

1ylnyy1xlnx=lnx+1\dfrac{1}{y\ln y}y'-\dfrac{1}{x\ln x}=\ln x+1

y=ylny(1xlnx+lnx+1)y'=y\ln y(\dfrac{1}{x\ln x}+\ln x+1)

y=xxxxxlnx(1xlnx+lnx+1)y'=x^{x^x}x^x\ln x(\dfrac{1}{x\ln x}+\ln x+1)

e.


y=(sinx)2xy = (\sin x)^{2x}

(lny)=(2xln(sinx))(\ln y)'=(2x\ln(\sin x))'

yy=2ln(sinx)+2x(cosxsinx)\dfrac{y'}{y}=2\ln(\sin x)+2x(\dfrac{\cos x}{\sin x})

y=2(sinx)2x(ln(sinx)+xcotx)y'=2(\sin x)^{2x}(\ln(\sin x)+x\cot x)

f)


(ln(x²+y²)+2arctan(y/x))=(1)(\ln(x²+y²) + 2\arctan(y/x))' = (1)'

2x+2yyx2+y2+21+y2/x2(yxyx2)=0\dfrac{2x+2yy'}{x^2+y^2}+\dfrac{2}{1+y^2/x^2}(\dfrac{y'}{x}-\dfrac{y}{x^2})=0

x+yy+xyy=0x+yy'+xy'-y=0

y=yxy+xy'=\dfrac{y-x}{y+x}

g)


((x+y)3)=((xy)2)((x+y)^3)' =( (x-y)^2)'

3(x+y)2(1+y)=2(xy)(1y)3(x+y)^2(1+y')=2(x-y)(1-y')

y=2(xy)3(x+y)23(x+y)2+2(xy)y'=\dfrac{2(x-y)-3(x+y)^2}{3(x+y)^2+2(x-y)}


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment