Question

Use the product rule to find the derivative of the given function.
y=(3 x³+4)(3 x-5)

Answer 1

To find the derivative of the function y=(3x³+4)(3x-5), we apply the product rule, which results in y’ = 36x³ - 45x² + 12.

Step-by-step explanation:

To find the derivative of the function y=(3x³+4)(3x-5) using the product rule, we will apply the rule which states that the derivative of a product of two functions is given by the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function.

Let’s define the two functions as follows:

• f(x) = 3x³+4
• g(x) = 3x-5

We have to find the derivatives f’(x) and g’(x) which are:

• f’(x) = derivative of 3x³ + 4 = 9x²
• g’(x) = derivative of 3x - 5 = 3

Using the product rule, we get:

y’ = f’(x)g(x) + f(x)g’(x)

Substituting the derivatives and functions, we get:

y’ = (9x²)(3x-5) + (3x³+4)(3)

Simplifying this, we obtain:

y’ = 27x³ - 45x² + 9x³ + 12

Combining like terms, we get the final derivative:

y’ = 36x³ - 45x² + 12