Final answer:
To find the partial derivatives of f(x,y,z)=ln(9x+7y+6z), we use the chain rule. fₓ = 9/(9x+7y+6z), fᵧ = 7/(9x+7y+6z), and f = 6/(9x+7y+6z).
Step-by-step explanation:
To find the partial derivatives, we need to compute the derivative of the function with respect to each variable separately. Let's start with fₓ:
fₓ = (∂f/∂x) = (∂/∂x) ln(9x+7y+6z)
To find this derivative, we need to use the chain rule. The derivative of ln(u) with respect to u is 1/u, and we have u = 9x+7y+6z. So, applying the chain rule, we get:
fₓ = 1/(9x+7y+6z) * (∂/∂x)(9x+7y+6z) = 1/(9x+7y+6z) * 9 = 9/(9x+7y+6z)
Similarly, we can find fᵧ and f:
fᵧ = (∂f/∂y) = (∂/∂y) ln(9x+7y+6z)
fᵧ = 1/(9x+7y+6z) * (∂/∂y)(9x+7y+6z) = 7/(9x+7y+6z)
f = (∂f/∂z) = (∂/∂z) ln(9x+7y+6z)
f = 1/(9x+7y+6z) * (∂/∂z)(9x+7y+6z) = 6/(9x+7y+6z)