Derivatives Product

Question

Use the product rule to find the derivative of (3x^9+7x^10)(4e^x+5)

Use e^x for . You do not need to expand out your answer.

I know product rulef（x）=g（x）*h（x），what is g(x) and h(x)?

Answer 1

Write f(x) = (3x⁹ + 7x¹⁰)(4e^x + 5).

For f(x) = g(x)h(x), Product Rule is f'(x) = g(x)h'(x) + g'(x)h(x).

Then write the derivative sought as f'(x) = (3x⁹ + 7x¹⁰) times (4e^x + 0) plus (27x⁸ + 70x⁹) times (4e^x + 5).

Simplify to f'(x) = 4e^x(3x⁹ + 7x¹⁰ + 27x⁸ + 70x⁹) + 5(27x⁸ + 70x⁹).

Rewrite as f'(x) = 4e^x(7x¹⁰ + 73x⁹+ 27x⁸) + 350x⁹ + 135x⁸.