Question

Given the equation y ln(x^2 + y^4 + 5) = 8, evaluate dy/dx. Assume that the equation implicitly defines y as a differentiable function of x. Choose the correct answer below. dy/dx = -2yx/(x^2 + y^4 + 5) ln (x^2 + y^4 + 5) + 4y^4 dy/dx = 8y ln (x^2 + y^4 + 5)/8(x^2 + y^4 + 5) + 4y^5 dy/dx = -x/2y^3 dy/dx = -2xy^2 - 4y^5/8 ln (x^2 + y^4 + 5)

Answer 1

`(dy)/(dx) = - (2\cdot x \cdot y)/((x^(2)+y^(4)+5)\cdot \ln(x^(2)+y^(4)+5)+4\cdot y^(4))`

Explanation:

The implicite derivative of the function is:

`(dy)/(dx)\cdot \ln (x^(2)+y^(4)+5) + (y)/((x^(2)+y^(4)+5))\cdot (2\cdot x + 4\cdot y^(3)\cdot (dy)/(dx) )=0`

`(dy)/(dx)\cdot(x^(2)+y^(4)+5)\cdot \ln (x^(2)+y^(4)+5)+y\cdot (2\cdot x+4\cdot y^(3)\cdot (dy)/(dx) )= 0`

`[(x^(2)+y^(4)+5)\cdot \ln(x^(2)+y^(4)+5)+4\cdot y^(4)]\cdot (dy)/(dx) + 2\cdot x \cdot y = 0`

`(dy)/(dx) = - (2\cdot x \cdot y)/((x^(2)+y^(4)+5)\cdot \ln(x^(2)+y^(4)+5)+4\cdot y^(4))`