Question

Use the product rule to find the derivative of

(10 x⁸-9 x⁴)(6 eˣ+3)
Use e∧x for eˣ. You do not need to expand out your answer.

Answer 1

To find the derivative of the function (10x⁸ - 9x⁴)(6eˣ + 3), we use the product rule, yielding (80x⁷ - 36x³)*(6eˣ + 3) + (10x⁸ - 9x⁴)*6eˣ.

Step-by-step explanation:

We need to use the product rule to find the derivative of the function (10x⁸ - 9x⁴)(6eˣ + 3). The product rule states that the derivative of the product of two functions u(x) and v(x) is u'(x)v(x) + u(x)v'(x), where u'(x) and v'(x) are the derivatives of u(x) and v(x), respectively.

Let's define our functions:

• u(x) = 10x⁸ - 9x⁴
• v(x) = 6eˣ + 3

Now, find the derivatives of each:

• u'(x) = ∂/∂x (10x⁸ - 9x⁴) = 80x⁷ - 36x³
• v'(x) = ∂/∂x (6eˣ + 3) = 6eˣ

Applying the product rule:

(u(x)v(x))' = u'(x)v(x) + u(x)v'(x)

Plug in the derivatives and original functions:

(10x⁸ - 9x⁴)(6eˣ + 3)' = (80x⁷ - 36x³)*(6eˣ + 3) + (10x⁸ - 9x⁴)*6eˣ

We do not need to expand the answer; we leave it in this factored form as requested by the student.