Final answer:
The derivative of the function y = (5x^3 + 4) (2x^7 − 6) is found using the product rule and simplifying to get dy/dx = 100x^9 + 56x^6 - 90x^2.
Step-by-step explanation:
To find the derivative of the function y = (5x^3 + 4) (2x^7 − 6), we'll use the product rule. The product rule 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 by u = 5x^3 + 4 and the second function by v = 2x^7 − 6. Then the derivative of u with respect to x is u' = 15x^2, and the derivative of v with respect to x is v' = 14x^6.
Applying the product rule, the derivative of y with respect to x is:
dy/dx = u'v + uv' = (15x^2)(2x^7 − 6) + (5x^3 + 4)(14x^6)
Simplifying this expression, we get:
dy/dx = 30x^9 - 90x^2 + 70x^9 + 56x^6
Combining like terms:
dy/dx = 100x^9 + 56x^6 - 90x^2