i need to use implicit differentiation to find the derivative:
yx+y = 8
i keep getting y'=-1-y.
i already know the derivative of this, by using explicit by simplification:
y(x+1)=8
y=(8)/(x+1)
y'= -(8)/(x+1)^2
how do i go about doing this the right way?
yx+y = 8
y dx/dx + x dy/dx + dy/dx = 0
(x+1)dy/dx = -y
dy/dx = -y/(x+1)
Oddly, the answers agree, since
8 = y(x+1), so
-8/(x+1)^2 = -(8/(x+1))/(x+1) = -y/(x+1)
To solve this equation using implicit differentiation, you need to follow these steps:
Step 1: Differentiate both sides of the equation with respect to x.
For the left side, you treat y as a function of x and use the product rule.
For the right side, the derivative of a constant (8 in this case) is 0.
Applying these steps to the equation yx + y = 8:
Differentiating the left side:
(d/dx)(yx) + (d/dx)(y)
Step 2: For the left side, we have two terms to differentiate.
Applying the product rule:
(d/dx)(yx) = y * (d/dx)(x) + x * (d/dx)(y)
= y * 1 + x * (d/dx)(y)
= y + x * (dy/dx)
Differentiating the right side:
(d/dx)(8) = 0
Substituting the derivatives back into the equation, we have:
y + x * (dy/dx) + (dy/dx) = 0
Step 3: Now, we need to isolate dy/dx by rearranging the equation:
(dy/dx) + x * (dy/dx) = -y
(dy/dx) * (1 + x) = -y
Step 4: Divide both sides of the equation by (1 + x):
(dy/dx) = -y / (1 + x)
Therefore, the derivative of the equation yx + y = 8 using implicit differentiation is dy/dx = -y / (1 + x).
It seems that your answer of y' = -1 - y is incorrect. However, when you simplify the equation by moving terms around, you arrive at the correct derivative: y' = -(8) / (x+1)^2.