SOLUTION: Let y be defined implicitly by the equation ln(5y)=3xy. Use implicit differentiation to find the second derivative of y with respect to x. d^2y/dx^2 = Simplify fully please!

Question 1200776: Let y be defined implicitly by the equation ln(5y)=3xy. Use implicit differentiation to find the second derivative of y with respect to x.
d^2y/dx^2 =
Simplify fully please!
Answer by asinus(45) (Show Source): You can put this solution on YOUR website!
**1. Differentiate Implicitly**
* **Original Equation:**
ln(5y) = 3xy
* **Differentiate both sides with respect to x:**
(1 / (5y)) * (5 * dy/dx) = 3y + 3x * dy/dx
* **Simplify:**
(1 / y) * dy/dx = 3y + 3x * dy/dx
* **Isolate dy/dx:**
(1 / y) * dy/dx - 3x * dy/dx = 3y
dy/dx * (1/y - 3x) = 3y
dy/dx = (3y) / (1/y - 3x)
dy/dx = (3y²) / (1 - 3xy)
**2. Differentiate Implicitly Again**
* **Differentiate both sides of dy/dx with respect to x:**
d²y/dx² = [(3 * 2y * dy/dx) * (1 - 3xy) - (3y²) * (-3y)] / (1 - 3xy)²
* **Substitute the expression for dy/dx:**
d²y/dx² = [(6y * (3y²) / (1 - 3xy)) * (1 - 3xy) - (3y²) * (-3y)] / (1 - 3xy)²
* **Simplify:**
d²y/dx² = [18y³ - (-9y³)] / (1 - 3xy)²
d²y/dx² = (27y³) / (1 - 3xy)²
**Therefore:**
**d²y/dx² = (27y³) / (1 - 3xy)²**
This is the second derivative of y with respect to x.

RELATED QUESTIONS
Suppose ln (x + 2y) = x^4. Use implicit differentiation to find the derivative of y... (answered by addingup)

Let {{{ e^(xy) = 4x+4y+20 }}} Use implicit differentiation to find the derivative of y (answered by Fombitz)

Find the derivative of the given equation using implicit differentiation. 1. x^2+y^2=x+y (answered by Solver92311)

Use logarithmic differentiation to find the derivative of the function. y = (ln... (answered by Fombitz)

Use logarithmic differentiation to find the derivative of the function. y=(ln(x+8))^x (answered by Solver92311)

implicit diffrention problem Use implicit differentiation to find an equation of the... (answered by Edwin McCravy)

use implicit differentiation to find an equation of the tangent line to the graph of... (answered by Alan3354)

Use implicit differentiation to find the equation of the tangent line at the given point (answered by Edwin McCravy)

Please help me with my assignment. Use Implicit differentiation to find the Derivative... (answered by ikleyn)