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

1 Answer
Aug 9, 2015

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)])