Final answer:
The derivative of the function y = (x+6)⁷ (x+3)⁵ is found by applying the product and power rules of differentiation, resulting in y' = 7(x+6)⁶(x+3)⁵ + (x+6)⁷ 5(x+3)⁴.
Step-by-step explanation:
To find the derivative of the function y = (x+6)⁷ (x+3)⁵, we need to apply the product rule combined with the power rule of differentiation. The product rule states that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. In our case:
Let u = (x+6)⁷ and v = (x+3)⁵. Therefore, the derivative u' is 7(x+6)⁶ and the derivative v' is 5(x+3)⁴.
Using the product rule, the derivative of y is:
y' = u'v + uv'
Substitute u, u', v, and v' into the equation:
y' = 7(x+6)⁶(x+3)⁵ + (x+6)⁷ · 5(x+3)⁴
This is the derivative of the given function.