f(x)=x^2/3 * (2-x) = x^2/3 * (2-x)

Question

find the 1st and 2nd derivative

Answer 1

Robert's answer looks good.  If, however, the 3(2-x) is in the denominator, then try this:

 

f(x) = x^2 / (6-3x)

 

Use Chain Rule where a = x^2 and b = (6-3x)^-1

 

f'(x) = ab' + ba'

f'(x) = (x^2)(-1)(-3)(6-3x)^-2 + 2x(6-3x)^-1

f'(x) = (3x^2)/(6-3x)^2 + (2x)/(6-3x)

 

Now multiply top and bottom of second term by (6-3x) and simplify: 

f'(x) = (3x^2)/(6-3x)^2 + (2x)(6-3x)/(6-3x)^2

 

Combine terms now that we have a common denominator:

f'(x) = (3x^2 + 12x - 6x^2) / (6-3x)^2

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

f'(x) = (3x(4-x)) / (9(2-x)^2)

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

 

For the second derivative, now factor out the (1/3) and use chain rule again where a = (4x-x^2) and b = (2-x)^-2

 

f'(x) = (1/3) [ x(4-x) / (2-x)^2 ]

f''(x) = (1/3) [ (4x-x^2)(-2)(-1)(2-x)^-3 + ((2-x)^-2)(4-2x) ]

 

Multiply terms and simplify:

f''(x) = (1/3) [ (8x-2x^2) / ((2-x)^3)  +  (4-2x) / ((2-x)^2)  ]

 

To simplify further, multiply top and bottom of the second term by (2-x) and combine terms:

f''(x) = 8 / (3(2-x)^3)

 

Happy Thanksgiving!

 

 

 

Answer 2

d/dx [x²/3(2-x)] 

 

I am going to assume STRICK order of operations and go left to right on the mult. and divide

If you meant something else comment with brackets everywhere to let us know.

 

rearrange to

d/dx [1/3 x²(2-x)]

 

now you have a constant, an x² term, and a (2-x)

factor out the constant

= 1/3 d/dx [(2-x)x²]

 

use the product rule on the last 2 terms

d/dx (uv) = v du/dx + u dv/dx

where u=2-x and v=x²  --or vice versa

 

=1/3[ x² d/dx (2-x) + (2-x) d/dx (x²) ]

 differentiate the sum term by term, factoring out the constants

 

=1/3 [ (2-x) d/dx(x²⁾ + (d/dx(2) - d/dx(x)) x² ]

 

Can you take it from here to get

 

f'x = d/dx [1/3 x²(2-x)] = 1/3 (4-3x) x

 

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the second derivative is the derivative of the first derivative

f''x = d/dx [f'x] = d/dx [1/3 (4-3x) x]

 

factor out the constant

= 1/3 d/dx [ (4-3x)x ]

 

use the product rule

then differentiate term by term factoring out constants

 

can you figure out the steps from here to get

 

f''x = 4/3 - 2x