Answer :
Final answer:
To find the derivative of the given function, the product rule of differentiation is applied. First, the derivatives of each part of the function are calculated, then the product rule is applied to combine these derivatives, finally simplifying the result.
Explanation:
The question asks to find the derivative and simplify the expression y = (2x⁷ + 5)(7x⁶ - 9x⁴ - 8). To solve this, we'll apply the product rule of differentiation, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Let's denote the first function as f(x) = 2x⁷ + 5 and the second function as g(x) = 7x⁶ - 9x⁴ - 8.
First, we find the derivatives: f'(x) = 14x⁶ and g'(x) = 42x⁵ - 36x³. Next, applying the product rule: y' = f'(x)g(x) + f(x)g'(x). After substitution and simplification, we get: y' = (14x⁶)(7x⁶ - 9x⁴ - 8) + (2x⁷ + 5)(42x⁵ - 36x³). This simplifies to the final derivative, showcasing how to apply the product rule and perform algebraic simplification.
