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

Question

Emma Aldridge · Accepted Answer

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

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)])#

Samantha Adkison · Answer

undefined To use the Product Rule to find the derivative of $ (7x^4 + 2x^6) \sin(7x) $, differentiate each term separately, then apply the Product Rule to combine them. The derivative is $ f'(x) = (28x^3 + 12x^5) \sin(7x) + (7x^4 + 2x^6) \cdot 7 \cos(7x) $.