What is the slope of xy^3+xy=10xy3+xy=10 at the point (5,1)(5,1)?

2 Answers
Jul 13, 2017

Slope =-1/10=110 at (5, 1)(5,1)

Explanation:

xy^3+xy=10xy3+xy=10

Applying implicit differentiation:

x*3y^2dy/dx+y^3*1 +xdy/dx+y*1 =0x3y2dydx+y31+xdydx+y1=0

(3xy^2+x)dy/dx = -(y^3+y)(3xy2+x)dydx=(y3+y)

:. dy/dx = (-(y^3+y))/(3xy^2+x)

dy/dx represents the slope of the tangent to the curve at any point (x_1, y_1) on the curve.

Slope of the tangent (m) at (5, 1)

= (-(1+1))/(3*5*1 +5) = (-2)/(15+5) = -2/20

=-1/10

Mar 30, 2018

"slope "=-1/10

Explanation:

"differentiate "color(blue)"implicitly with respect to x"

"differentiate "xy^3" and "xy" using the "color(blue)"product rule"

#(x.3y^2dy/dx+y^3 .1)+(xdy/dx+y.1)=0#

rArr3xy^2dy/dx+y^3+xdy/dx+y=0

rArrdy/dx(3xy^2+x)=-y^3-y

rArrdy/dx=-(y^3+y)/(3xy^2+x)

"at "(5,1)

dy/dx=-(1+1)/(15+5)=-2/20=-1/10larrcolor(red)"slope"