Question

How do you use the Product Rule to find the derivative of `(7x^4 + 2x^6) sin(7x)`?

Answer 1

`y^' = x^3 * [4 * (3x^2 + 7) * sin(7x) + 7x * (2x^2 + 7) * cos(7x)]`

Explanation:

The product rule allows you to differentiate functions that take the form

`y = f(x) * g(x)`

by using the formula

`color(blue)(d/dx(y) = [d/dx(f(x))] * g(x) + f(x) * d/dx(g(x)))`

In your case, you can think of the function as being

`y = underbrace((2x^6 + 7x^4))_(color(blue)(f(x))) * underbrace(sin(7x))_(color(green)(g(x)))`

This means that you can write

`d/dx(y) = [d/dx(2x^6 + 7x^4)] * sin(7x) + (2x^6 + 7x^4) * d/dx(sin(7x))`

To differentiate `sin(7x)`, you're going to use the chain rule for `sin u`, with `u = 7x`

`d/dx(sinu) = [d/(du) * (sinu)] * d/dx(u)`

`d/dx(sinu) = cosu * d/dx(7x)`

`d/dx(sin(7x)) = cos(7x) * 7`

This means that your target derivative will be

`y^' = (12x^5 + 28x^3) * sin(7x) + (2x^6 + 7x^4) * 7cos(7x)`

`y^' = 4x^3 * (3x^2 + 7) * sin(7x) + 7x^4 * (2x^2 + 7) * cos(7x)`

`y^' = color(green)(x^3 * [4 * (3x^2 + 7) * sin(7x) + 7x * (2x^2 + 7) * cos(7x)])`