Question

Let
`f(x)=(2x+7)^3` and
`g(x)=cos^2(4x)`. Find

(a)
`f'(x)`

(b)
`g'(x)`

Answer 1

f(x) = (2x + 7)³
f(x) = (2x + 7)(2x + 7)(2x + 7)
f(x) = [2x(2x + 7) + 7(2x + 7)](2x + 7)
f(x) = [2x(2x) + 2x(7) + 7(2x) + 7(7)](2x + 7)
f(x) = (4x² + 14x + 14x + 49)(2x + 7)
f(x) = (4x² + 28x + 49)(2x + 7)
f(x) = 4x²(2x + 7) + 28x(2x + 7) + 49(2x + 7)
f(x) = 4x²(2x) + 4x²(7) + 28x(2x) + 28x(7) + 49(2x) + 49(7)
f(x) = 8x³ + 28x² + 56x² + 196x + 98x + 343
f(x) = 8x³ + 84x² + 294x + 343


`f'(x) = (f(x + \delta x) - f(x))/(\delta x)`

`f'(x) = \frac{{([(8x^(3) + 24dx^(3) + 24d^(2)x^(3) + 3d^(3)x^(3))] + [84d^(2)x^(2) + 162dx^(2) + 84x^(2)] + [294x + 294dx] + 343}) - (8x^(3) + 84x^(2) + 294x + 343)}{dx}`

`f'(x) = ((8x^(3) - 8x^(3)) + (84x^(2) - 84x^(2)) + (294x - 294x) + (343 - 343) + 24dx^(3) + 24d^(2)x^(3) + 3d^(3)x^(3) + 84d^(2)x^(2) + 162dx^(2) + 294dx)/(dx)`

`f'(x) = (24dx^(3) + 24d^(2)x^(3) + 3d^(3)x^(3) + 84d^(2)x^(2) + 162dx^(2) + 294dx)/(dx)`

`f'(x) = 24x^(2) + 24dx^(2) + 3d^(2)x^(2) + 84dx + 162x + 294`

`f'(x) = 24x^(2) + 162x + 294`
---------------------------------------------------------------------------------------------------------------
g(x) = cos²(4x)
g(x) = cos(4x)cos(4x)


`g'(x) = D\{cos(4x)\}cos(4x) + cos(4x)D\{cos(4x)\}`

`g'(x) = -4sin(4x)cos(4x) - 4sin(4x)cos(4x)`

`g'(x) = -8sin(4x)cos(4x)`

`g'(x) = -8[2sin(2x)cos(2x)][cos^(2)(2x) - sin^(2)(2x)]`

`g'(x) = -8[4sin(x)cos^(3)(x) - 4sin^(3)(x)cos(x)][cos^(4)(x) - 2sin^(2)(x)cos^(2)(x) + sin^(4)(x) - 4sin(x)cos(x)`

`g'(x) = -8[4sin(x)cos^(7)(x) - 8sin^(3)(x)cos^(5)(x) - 16sin^(2)(x)cos^(4)(x) + 4sin^(5)(x)cos^(3)(x) - 4sin^(3)(x)cos^(5)(x) + 8sin^(5)(x)cos^(3)(x) + 16sin^(4)(x)cos^(2)(x) + 4sin^(7)(x)cos^(3)(x)}]`

`g'(x) = -32sin(x)cos^(7)(x) + 64sin^(3)cos^(5)(x) + 128sin^(2)(x)cos^(4)(x) - 32sin^(5)(x)cos^(3)(x) + 32sin^(3)(x)cos^(5)(x) - 64sin^(5)(x)cos^(3)(x) - 128sin^(4)(x)cos^(2)(x) + 32sin^(7)cos^(3)(x)`

Answer 2

`f'(x)=6(2x+7)^(2)`

`g'(x)=-8*cos(4x)*sin(4x)`