how to find the first and second derivatives of

Question

x²(4X+1)⁴

f'(x)=

f''(x)=

Answer 1

The basic problem is two things being multiplied together meaning the basic derivative property to use is the product rule:

(u•v)' = u'v + uv'

u = x²

u' (using the power rule) is 2x

v = (4x + 1)⁴

To find v' we must use both the power rule AND the chain rule so v' = 4(4x + 1)³•(4) (the "4" at the end is the derivative of the "4x + 1")

Put these together using (u•v)' = u'v + uv' = (2x)(4x + 1)⁴ + x²[4(4x + 1)³•(4)] and then simplify

Answer 2

g(x) = x²

h(x) = 4x + 1

k(x) = x⁴

g(x) * (k(h(x))

Use Chain Rule for derivative of (k(h(x))

Apply Product Rule to multiplhy by derivative of g(x)