Final answer:
To differentiate the function y = (x³ + 7x - 1)(5x + 2), apply the product rule to find y' = 20x³ + 6x² + 70x + 9 after distributing and combining like terms.
Step-by-step explanation:
To differentiate the function y = (x³ + 7x - 1)(5x + 2), we need to apply the product rule, since this is a product of two functions of x. The product rule states that the derivative of a product uv is u'v + uv', where u' and v' are the derivatives of u and v respectively.
First, we identify u = x³ + 7x - 1 and v = 5x + 2. Their derivatives are u' = 3x² + 7 and v' = 5. Using the product rule, the derivative of y is:
y' = (3x² + 7)(5x + 2) + (x³ + 7x - 1)(5)
Now we simply distribute and combine like terms to get the final answer:
y' = 15x³ + 6x² + 35x + 14 + 5x³ + 35x - 5
y' = 20x³ + 6x² + 70x + 9
This is the simplified form of the derivative of the given function.