Question

Let f(x) = 3x2 + x - 3 and g(x) = x2 - 5x + 1. Find f(x) - g(x).

2x2 - 4x - 2

2x2 - 4x - 4

2x2 + 6x - 2

2x2 + 6x - 4

Answer 1

Let’s re-write the function as follows:

y = (x2 + x)5(1 + x3)2
Let’s consider a function k as follows:

k = x2 + x
k' = 2x + 1
Let’s consider a function h as follows:

u = k5
= (x2 + x)5

u' = nkn-1k'
= 5(x2 + x)4(2x + 1)
= 5(2x + 1)(x2 + x)4
Let’s consider a function g as follows:

g = 1 + x3

g’ = 3x2
Let’s consider a function m as follows:

v = g2
= (1 + x3)2

v' = ngn-1g'
= 2(1 + x3)(3x2)
= 6x2(1 + x3)
Let’s consider a function f as follows:

f = uv
To calculate the derivative, we use the formula (uv)' = u'v + uv':

f' = u'v + uv'
= 5(2x + 1)(x2 + x)4(1 + x3)2 +(x2 + x)56x2(1 + x3)
= 5(2x + 1)(x2 + x)4(1 + x3)2 +6x2(1 + x3)(x2 + x)5
= (x2 + x)4(1 + x3)[5(2x + 1)(1 + x3) + 6x2(x2 + x)]