Answer to Question #182539 in Calculus for Boris Iradukunda

Question

Let y be defined implicitly by the equation ln(3y)=5xy Use implicit differentiation to find the first derivative of y with respect to x

a)dy/dx=

Use implicit differentiation to find the second derivative of y with respect to x

b)d2y/dx2= 

c)Find the point on the curve where d2y/dx2=0

Answer 1

ln(3y) = 5xy

(a) Differentiating with respect to x,

(dy/dx)*1/y = 5(y+ xdy/dx)

By transposition we have,

dy/dx = (5y²)/(1 - 5xy) ....................(1)

(b) Again differentiating equation (1) with respect to x,

d²y/dx² = { 10y dy/dx (1- 5xy) + 5(y+ xdy/dx)*(5y²) } / ( 1 - 5xy)²

On further solving we have,

d²y/dx² = { 25y³ * (3 -10xy) } / ( 1- 5xy)³ ..........(2)

(c) For the required point we set equation (2) equal to 0,

Hence, d²y/dx² = 0

y=0 or xy = 0.3

but y = 0 is not possible

So xy=0.3

substituting the value of xy in the curve we have

y = 1.493

x = 0.201