Finding the second derivitive?

Question

Im having some trouble with my calculus homework. Our teacher gave us a worksheet of some problems, and I got stuck on this second derivitive.

the Origional problem was - f(x)=(1/5)(x^5 - 5)^3

I used the chain rule and got - f'(x)=(3/5)(x^5 - 5)^2 * 5x^4

But when I went to do the second derivitive.... I got lost because its the chain rule and product rule right? can someone show me how to do this...?

Would the second derivitive be

(6/5)(x^5 - 5) * 5x^4 * 20x^3

That one guy had like some other stuff after this.... I dont understand why he double chain ruled like that...

Answer 1

f'(x) = (3/5)(x^5 - 5)^2 (5x^4)

The best thing to do is simplify this as much as you can prior to taking the second derivative. Notice how the 5 in front of the x^4 can be merged with (3/5) to make just 3. In fact, let's move 5x^4 to the front altogether.

f'(x) = (3/5)5x^4 (x^5 - 5)^2
f'(x) = 3x^4 (x^5 - 5)^2

Now, use the product rule.

f''(x) = [12x^3] (x^5 - 5)^2 + (3x^4) (2)(x^5 - 5) (5x^4)

We can even simplify this.

f''(x) = [12x^3] (x^5 - 5)^2 + (30x^8)(x^5 - 5)

Factoring out 6x^3(x^5 - 5),

f''(x) = 6x^3(x^5 - 5) [2(x^5 - 5) + 5x^5]
f''(x) = 6x^3(x^5 - 5) [2x^5 - 10 + 5x^5]
f''(x) = 6x^3(x^5 - 5) [7x^5 - 10]

Answer 2

you're right on the first derivative

f''(x) = (6/5)(x^2 - 5)*5x^4 +20x^3 * (3/5)(x^5 - 5)^2

double chain rule. f = uv
f' = uv' + vu'

Answer 3

f(x)=1/5(x^5-5)^3
f'(x)=3/5(x^5-5)^2*5x^4
form: f(x)'=K(U^n)*(V^n) f(x)''=?
we know that:
(U^n)'=nU'*U^n-1 and
(U^n*V^n)'=(nU'*U^n-1)V^n+(nV'*V^n-1)U^n
U^n=(x^5-5)^2 ; (U^n)'=2*5x^4(x^5-5)^2-1
V^n=5x^4 ; (V^n)'=4*5x^3
K =3/5
f(x)''=K[(nU'*U^n-1)(V^n)+(nV'*V^n-1)U^n]
let's replace the letters...
f(x)''=3/5[10x^4(x^5-5)(5x^4)+(20x^3)(x^5-5)^2
f(x)''=3/5[(50x^8)(x^5-5)+(20x^3)(x^5-5)^2]
let's simplify:
f(x)''=30x^8(x^5-5)+12x^3(x^5-5)^2
factorise by 6x^3(x^5-5)
f(x)''=6x^3(x^5-5)[5x^5+2(x^5-5)]

Answer 4

simplify f'(x) to 3x^4(x^5-5)^2

f''(x) = 3 * 4x^3(x^5-5)^2 + 2(x^5-5)(5x^4)(3x^4)
f''(x)=12x^3(x^5-5)^2 + 30(x^8)(x^5-5)