I need help simplifying this problem.

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

f'(x)=x^4/5*2(x-4)+(x-4)^2*4/5x^-1/5

it is suposed to end up as this:

1/5x^-1/5(x-4)[5*x*2+(x-4)*4]

but how do i get it to there? I need to see all the steps and how to get them. Thanks

your first line is good

now, do you notice x^(-1/5) and x^(4/5) ?

isn't x^(-1.5) the highest common factor ? (just like a^4 would be the HCF for a^4 and a^6)

also (x-4) is a common factor,
as well as -1/5 (pretend your first term was (5/5)x^4/5*2(x-4)

so
(-1/5)(x^(-1/5)(x-4)[10 + 4(x-4)]

= (-1/5)(x^(-1/5)(x-4)[4x-6]
= (-2/5)(x^(-1/5)(x-4)[2x-3]

prove that sin invers (1/5)+COT INVERSE (3)=0

To simplify the expression and arrive at the given result, we'll break down the process step-by-step:

1. Start with the given derivative: f'(x) = (x^(4/5) * 2(x-4)) / 5 + (x-4)^2 * 4/(5x^(1/5))

2. Simplify the first term:
(x^(4/5) * 2(x-4)) / 5 = (2x^(9/5) - 8x^(4/5)) / 5

3. Simplify the second term:
(x-4)^2 * 4/(5x^(1/5)) = 4(x-4)^2 / (5x^(1/5))

4. Multiply the first term by x^(1/5)/x^(1/5) to rationalize the denominator:
[(2x^(9/5) - 8x^(4/5)) / 5] * (x^(1/5)/x^(1/5)) = (2x^(14/5) - 8x^(9/5)) / (5x^(1/5))

5. Multiply the second term by x^(-1/5)/x^(-1/5) to rationalize the denominator:
[4(x-4)^2 / (5x^(1/5))] * (x^(-1/5)/x^(-1/5)) = 4(x-4)^2 x^(-1/5) / (5x^(1/5))

6. Combine the terms to get a common denominator:
(2x^(14/5) - 8x^(9/5)) / (5x^(1/5)) + 4(x-4)^2 x^(-1/5) / (5x^(1/5))

7. Combine the numerators over the common denominator:
[2x^(14/5) - 8x^(9/5) + 4(x-4)^2 x^(-1/5)] / (5x^(1/5))

8. Factor out the common factor of x^(-1/5) from the numerator:
2x^(14/5) - 8x^(9/5) + 4(x-4)^2 x^(-1/5) = x^(-1/5) * (2x^(15/5) - 8x^(10/5) + 4(x-4)^2)

9. Simplify further in the numerator:
x^(-1/5) * (2x^(15/5) - 8x^(10/5) + 4(x-4)^2) = x^(-1/5) * (2x^3 - 8x^2 + 4(x-4)^2)

10. Simplify (x-4)^2 to (x-4)(x-4) and distribute the 4:
x^(-1/5) * (2x^3 - 8x^2 + 4(x-4)(x-4))

11. Expand and simplify the expression:
x^(-1/5) * (2x^3 - 8x^2 + 4x^2 - 32x + 64)

12. Combine like terms:
x^(-1/5) * (2x^3 - 4x^2 - 32x + 64)

13. Finally, rewrite the expression factoring out the common factor of 1/5x^(-1/5):
(1/5x^(-1/5))(5 * 2x^3 - 5 * 4x^2 - 5 * 32x + 5 * 64)

14. Simplify the result:
(1/5x^(-1/5))(10x^3 - 20x^2 - 160x + 320)

And that's how you reach the given expression: 1/5x^(-1/5)(10x^3 - 20x^2 - 160x + 320).