Question

given f(x)=3x(x-1)^5, compute f'''(x)

Answer 1

Solution. Find the first derivative using the formula

$$(uv)'=u'v+v'u.$$

Let u=3x and v=(x-1)^5. Therefore get

$$f'(x)=(3x(x-1)^5)'=3(x-1)^5+3x*5(x-1)^4$$

Simplifying the expression we get

$$f'(x)=(3x-3)(x-1)^4+15x(x-1)^4=(18x-3)(x-1)^4$$

Let u=18x-3 and v=(x-1)^4. Therefore get

$$f''(x)=18(x-1)^4+(18x-3)*4(x-1)^3$$

Simplifying the expression we get

$$f''(x)=(18x-18)(x-1)^3+(72x-12)(x-1)^3$$

$$f''(x)=(90x-30)(x-1)^3$$

Let u=90x-30 and v=(x-1)^3. Therefore get

$$f'''(x)=90(x-1)^3+(90x-30)*3(x-1)^2$$

Simplifying the expression we get

$$f'''(x)=(90x-90)(x-1)^2+(270x-90)(x-1)^2=(360x-180)(x-1)^2$$

Answer.

$$f''(x)=(360x-180)(x-1)^2$$