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Find the derivative of the function y = (x+6)⁷ (x+3)⁵.

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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.

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