Question

Let y^4+5x=11y 4 +5x=11y, start superscript, 4, end superscript, plus, 5, x, equals, 11. What is the value of \dfrac{d^2y}{dx^2} dx 2 d 2 y ​ start fraction, d, squared, y, divided by, d, x, squared, end fraction at the point (2,1)(2,1)left parenthesis, 2, comma, 1, right parenthesis?

Answer 1

`(d^2y)/(dx^2) = (-300y^2)/((4y^3-11)^3)`

Explanation:

Given the expression
`y^4+5x=11y`, we are to find the second derivative
`(d^2y)/(dx^2)`

This differentiation will be implicit (indirect) as shown:

`4y^3 (dy)/(dx) + 5 = 11(dy)/(dx) \\4y^3 (dy)/(dx) - 11 (dy)/(dx) = -5\\(dy)/(dx)(4y^3-11) = -5\\(dy)/(dx) = (-5)/(4y^3-11)`

Differentiating the second time using quotient rule:

`(d^2y)/(dx^2) = (4y^3-11 (0)- (-5)12y^2(dy)/(dx) )/((4y^3-11)^2) \\\\(d^2y)/(dx^2) = (60y^2(dy)/(dx) )/((4y^3-11)^2) \\\\(d^2y)/(dx^2) = (60y^2((-5)/(4y^3-11) ) )/((4y^3-11)^2)\\\\(d^2y)/(dx^2) = (-300y^2)/((4y^3-11)^3)`