Final answer:
The derivative of the function z = e^(8x) * 2y with respect to x is found using the product rule, resulting in dz/dx = 8e^(8x) * 2y + e^(8x) * d(2y)/dx. This calculation assumes y as a function of x or uses partial derivatives.
Step-by-step explanation:
The student is asking to find the derivative of the function z = e^(8x) * 2y. Since z is a function of two variables, x and y, we will assume that y is a function of x, or the context is that of partial derivatives. However, since the question doesn't specify, we'll approach it as if taking a total derivative with respect to x, treating y as an implicit function of x. The derivative of a product of functions is given by the product rule, which states that the derivative of u*v, where u and v are functions of x, is u'v + uv'. Applying the product rule and the chain rule for the exponential function, the derivative dz/dx would be:
dz/dx = 8e^(8x) * 2y + e^(8x) * d(2y)/dx
We use the constant coefficient rule to move the constant 2 outside the derivative, and then apply the product rule again if we need to find d(2y)/dx explicitly. The derivative involves not only differentiation with respect to x but also takes into account the potential change in y with respect to x.