Question

Find the first partial derivatives of the function w = ln(x²y⁶z)?

Answer 1

To find the first partial derivatives of the function w = ln(x²y⁶z), we can differentiate the function with respect to each variable separately while treating the other variables as constants.

Step-by-step explanation:

To find the first partial derivatives of the function w = ln(x²y⁶z), we will differentiate the function with respect to each variable separately while treating the other variables as constants.

1. To find ∂w/∂x, we treat y and z as constants and differentiate ln(x²y⁶z) with respect to x using the chain rule. ∂w/∂x = 2xy⁶z/x² = 2y⁶z/x
2. To find ∂w/∂y, we treat x and z as constants and differentiate ln(x²y⁶z) with respect to y using the chain rule. ∂w/∂y = 6x²y⁵z/y = 6x²y⁴z
3. To find ∂w/∂z, we treat x and y as constants and differentiate ln(x²y⁶z) with respect to z using the chain rule. ∂w/∂z = x²y⁶z/z = x²y⁶