Question

Find dy / dx by implicit differentiation. xy + 5x + 5x2 = 4

Answer 1

HI,

Step-by-step explanation:

`xy+5x+5x^2=4\\\\y+xy'+5+10x=0\\\\y'=(dy)/(dx) =(-10x-5-y)/(x)`

Answer 2

( - y - 10x - 5)/x

Step-by-step explanation:

The question is asking for the derivative of the equation xy + 5x + 5x² = 4 using implicit differentiation. Implicit differentiation allows us to take derivatives without having to solve the equation for one variable. The derivative of the equation, using product rule and chain rule where needed, therefore becomes:

⇒ xy + 5x + 5x² = 4

⇒ d/dx[xy + 5x + 5x²] = d/dx[4]

⇒ d/dx[xy] + d/dx[5x] + d/dx[5x²] = 0

⇒ (x · dy/dx + y) + 5 + 10x = 0

⇒ x · dy/dx + y + 5 + 10x = 0

Solve for 'dy/dx':

⇒ x · dy/dx = - y - 10x - 5

∴ dy/dx = (- y - 10x - 5)/x

`\hrulefill`

Additional Information:

Product rule: d/dx[f(x)g(x)] = f(x)g'(x) + f'(x)g(x)

Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)