Please verify my answers and show me where I went wrong

[CHiMER4]

y=(8x^3+4x^2-2x-1)/2x+1
dy/dx=(24x^2+8x-2)/2

Dx(5x^5+9ax^2/3x^2)
=(25x^4+18ax)/6x

d/dx((5√x^3)-(3π/√x)
=((x)^3/5) - (3π/(x)^1/2)

NB that product/quotient rule should not be used to solve these

 

[Deleted member 4993]

y=(8x^3+4x^2-2x-1)/2x+1
dy/dx=(24x^2+8x-2)/2

NB that product/quotient rule should not be used to solve these

If the function is:

y=(8x^3+4x^2-2x-1)/(2x+1) ........Those parentheses are SUPER important.

If use of product/quotient rules are not allowed, one other way is to factorize the numerator (using long polynomial or synthetic division)and HOPE that the numerator has the denominator as a factor (thus the denominator will "cancel" out).

As it happens, for the first problem the denominator do cancel out.

factorize and continue....

 

[CHiMER4]

Okay
I did ask you said, and for the first question I got a derivative of 8x
3rd=3/(5x^(2/5)) + 3π/(2x^(3/2))

But how would I do this one:
Dx[((5x^5)+(9ax^2))/(3x^2)] where a is an unknown constant

 

[Deleted member 4993]

3rd=3/(5x^(2/5)) + 3π/(2x^(3/2))

d/dx((5√x^3)-(3π/√x)
=((x)^3/5) - (3π/(x)^1/2)

If you are saying:

\(\displaystyle \frac{d}{dx}\left[5\sqrt{x^3} - 3\frac{\pi}{\sqrt{x}}\right] \ \ ?=\ \ ? \frac{3}{5x^{\frac{2}{5}}} + 3\frac{\pi}{2x^{\frac{3}{2}}}\)

Then it is incorrect.

Please share your work in detail to correct your mistake/s.

 

[Otis]

… But how would I [proceed when] a is an unknown constant

Hi CHiMER4. Start the same way as the others: simplify the given expression.

As far as differentiating, simply treat terms that contain symbol a as constants.

For examples, the derivative of expressions like 1/2*sqrt(a) or 7a^2 or 51a/1000 is zero because each of those expressions represent a value that's constant.

?

 

[HallsofIvy]

y=(8x^3+4x^2-2x-1)/2x+1
dy/dx=(24x^2+8x-2)/2

Dx(5x^5+9ax^2/3x^2)
=(25x^4+18ax)/6x

d/dx((5√x^3)-(3π/√x)
=((x)^3/5) - (3π/(x)^1/2)
NB that product/quotient rule should not be used to solve these

I think you have misunderstood that "NB". You appear to have unterpreted i to mean you should take d(f/g)/dx to be (df/dx)/(dg/dx). No, the fact that you "should not use" those rules does not mean they do not apply.

What they mean is that you should write 5x^5+9ax^2/3x^2 as (5/3)x^3+ 3a and differentiate that.

Similarly write \(\displaystyle 5\sqrt{x^3}- 3\pi\sqrt{x}\) as \(\displaystyle 5x^{3/2}+ 3\pi x^{-1/2}\) and differentiate that.

It is not clear whether the first one is \(\displaystyle \frac{8x^3+ 4x^2- 2x- 1}{2x}+ 1\) or \(\displaystyle \frac{8x^3+ 4x^2- 2x- 1}{2x+ 1}\).
If it is the first then write it as \(\displaystyle 4x^2+ 2x- 1- (1/2)x^{-1}+ 1\).
If it is the second then you need to divide by 2x+ 1:
\(\displaystyle \frac{8x^3+ 4x^2- 2x- 1}{2x+ 1}= 4x^2- 1\). (The fact that it divides evenly makes me think this is the correct one!)