Differential Calculus: Find f'(x) if f(x) = 5 * 4^x - 2x^4 ln 4?

Question

The asterisk is a multiplication sign. the ln4 is just there with no sign next to it, im having trouble understanding the concepts so I would like to see the steps one would take in order to solve this and maybe possible some brief reasoning would be nice as well. they give me the following possible answers. Thanks in advance for any help.

1. f'(x) = (5 * (4^x) - 8x^3) ln 4
2. f'(x) = 5x * (4^x-1) - 2x^3 ln 4
3. f'(x) = (5 * (4^x) - 2x^3) ln 4
4. f'(x) = 20 * (4^x-1) - 8x^3 ln 4
5. f'(x) = (5 * (4^x-1) - 8x^4) ln 4
6. f'(x) = 5 * (4^x-1) - 2x^3 ln 4

Answer 1

f'(x) = (ln 4)(4^x)(5) - (8x^3)(ln 4)
f'(x) = ln 4 [(4^x)(5) - (8x^3)]

Ideas: Apply product rule and sum rule of derivatives.

f(x) = a^x
f'(x) = ln a a^x, only if a is a constant

Answer 2

Let3041ecfe6d67c6f8bcc34f5d5a65df89f( lnx ) lnx = lnx [3041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df89]^53041ecfe6d67c6f8bcc34f5d5a65df89f3041ecfe6d67c6f8bcc34f5d5a65df89?(3041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df89) lnx =5(a million/ lnx )((3041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df89)^4) lnx = lnx (5(3041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df89)^4)/3041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df89f?(e3041ecfe6d67c6f8bcc34f5d5a65df89^2) lnx =(5(3041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df89(e^2)^4)) lnx / lnx (e^2) lnx = lnx (5(2^4)) lnx / lnx (e^2) lnx =3041ecfe6d67c6f8bcc34f5d5a65df89803041ecfe6d67c6f8bcc34f5d5a65df89/3041ecfe6d67c6f8bcc34f5d5a65df89e^23041ecfe6d67c6f8bcc34f5d5a65df89Let3041ecfe6d67c6f8bcc34f5d5a65df89f( lnx ) lnx =3041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df89( lnx ^4)3041ecfe6d67c6f8bcc34f5d5a65df89f3041ecfe6d67c6f8bcc34f5d5a65df89?(3041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df89) lnx =(43041ecfe6d67c6f8bcc34f5d5a65df89^3) lnx /3041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df89^43041ecfe6d67c6f8bcc34f5d5a65df89=3041ecfe6d67c6f8bcc34f5d5a65df894/3041ecfe6d67c6f8bcc34f5d5a65df893041ecfe6d67c6f8bcc34f5d5a65df89f3041ecfe6d67c6f8bcc34f5d5a65df89?(3041ecfe6d67c6f8bcc34f5d5a65df89e^33041ecfe6d67c6f8bcc34f5d5a65df89) lnx =3041ecfe6d67c6f8bcc34f5d5a65df8943041ecfe6d67c6f8bcc34f5d5a65df89/ lnx (e^3)

Answer 3

I assume the problem is y = 5*4^(x-2) * x^(4ln4). If so, the product rule should work just fine:
dy/dx = 5*4^(x-2)* 4ln4x ^(4ln4-1)+x^(4ln4) * 5*ln4*4^(x-2)

Answer 4

d/dx (f + g) = d/dx (f) + d/dx (g)
d/dx (a^x) = a^x * ln(a)
d/dx (u^n) = n * u^(n-1)

f'(x) = d/dx (5*4^x) - d/dx (2x^4 * ln(4))

use the rules above
f'(x) = 5 (4^x) ln(4) - 2ln(4) * 4x^3

simplify
f'(x) = 5ln(4) 4^x - 8ln(4) x^3 <== answer