Final answer:
The derivative of y = x⁸e⁸ is found using the product rule, resulting in dy/dx = 8x⁷e⁸.
Step-by-step explanation:
The original function provided is y = x⁸e⁸. To differentiate this function with respect to x, we must apply the product rule, which is used when we have the product of two functions. The product rule states that for two functions u(x) and v(x), the derivative of their product u(x)v(x) is u'(x)v(x) + u(x)v'(x).
Let u(x) = x⁸ and v(x) = e⁸. The derivative of u(x) with respect to x is 8x⁷, and the derivative of v(x), which is just a constant, is 0 since the derivative of any constant is zero.
Applying the product rule:
dy/dx = u'(x)v(x) + u(x)v'(x)
dy/dx = 8x⁷e⁸ + x⁸(0)
dy/dx = 8x⁷e⁸