Suppose the equation F(x,y,z) = 𝑥^2 +𝑦^2 − 2𝑧^2 +12x −8z – 4 = 0 can be solved for z as z = f(x,y). Find 𝜕z/𝜕x at the point P(1,1,1). Use implicit dif f differentiation.
Why are you taking the derivative of both z and x at the same time? Are you making a mistake? I thought the problem meant partial derivative of z over partial derivative of x then plug in and solve. Am I right?
To find ∂z/∂x using implicit differentiation, we differentiate both sides of the equation with respect to x.
Starting with the given equation:
F(x, y, z) = 𝑥^2 + 𝑦^2 − 2𝑧^2 + 12x − 8z – 4 = 0
Differentiating both sides with respect to x:
∂F/∂x = ∂/∂x (𝑥^2 + 𝑦^2 − 2𝑧^2 + 12x − 8z – 4) = 2x + 12 - 8 ∂z/∂x
Since we are interested in finding ∂z/∂x, we can isolate this term:
2x + 12 - 8 ∂z/∂x = 0
Now we can solve for ∂z/∂x:
-8 ∂z/∂x = -2x - 12
∂z/∂x = (-2x - 12) / -8
To find ∂z/∂x at the point P(1, 1, 1), substitute x = 1 into the equation:
∂z/∂x = (-2(1) - 12) / -8
∂z/∂x = (-2 - 12) / -8
∂z/∂x = -14 / -8
∂z/∂x = 7 / 4
Therefore, ∂z/∂x at the point P(1, 1, 1) is 7/4.
what's the trouble?
2x - 4z 𝜕z/𝜕x + 12 - 8 𝜕z/𝜕x = 0
Now just solve for 𝜕z/𝜕x