How do you use the Product Rule to find the derivative of #g(x) = (2x^2 + 4x - 3) ( 5x^3 + 2x + 2)#?

1 Answer
Aug 16, 2015

Answer:

#g^' = 50x^4 + 80x^3 - 33x^2 + 24x + 2#

Explanation:

Notice that your function can be written as the product of two other functions

#g(x) = underbrace((2x^2 + 4x - 3))_(color(orange)(f(x))) * underbrace((5x^3 + 2x + 2))_(color(purple)(h(x)))#

This means that you can rule the product rule to get

#color(blue)(d/dx(g(x)) = [d/dx(f(x))] * h(x) + f(x) * d/dx(h(x)))#

Using this formula will get you

#d/dx(g(x)) = [d/dx(2x^2 + 4x - 3)] * (5x^3 + 2x + 2) + (2x^2 + 4x - 3) * d/dx(5x^3 + 2x + 2)#

#g^' = (4x + 4) * (5x^3 + 2x + 2) + (2x^2 + 4x - 3) * (15x^2 + 2)#

#g^' = 20x^4 + 8x^2 + 8x + 20x^3 + 8x + 8 + 30x^4 + 4x^2 + 60x^3 + 8x -45x^2 - 6#

#g^' = color(green)(50x^4 + 80x^3 - 33x^2 + 24x + 2)#