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Calculus/Derivative problem

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Question
Hello,

I'm having a hard time figuring out which rules to use on this function to take the derivative. (i.e. quotient rule, product rule, etc.) I was wondering if you could help me dissect the function to interpret how to take the derivative.

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

Thanks you very much.

Nick

Answer
Hello,

I'm having a hard time figuring out which rules to use on this function to take the derivative. (i.e. quotient rule, product rule, etc.) I was wondering if you could help me dissect the function to interpret how to take the derivative.

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

Thanks you very much.

Nick
----------------------------------
An algebraic expression is an indicated sequence of operations.  The 'type' of expression is determined by the LAST operation.

If the last operation is multiplication, it's a PRODUCT.
If the last operation is division, it's a QUOTIENT.
If the last operation is addition/subtraction, it's a SUM.
If the last operation is raising to a power(*), it's a POWER.

(*) raising to a power is called involution, but that term was obsolete by the beginning of the LAST century.
.................................

f(x) = 3 - [(8 + 3x)]^2/3 is a SUM, because if I give you  x = 47.928, or some such value, you:

Multiply by 3.
Add 8
Raise to the 2/3 power.
Subtract that from 3.

So you 'diff' each term separately --  the sum rule.
...............
Now the second term, I will call:  f1(x) = [(8 + 3x)]^2/3 is a POWER, because you

Multiply x by 3.
Add 8
Raise to the 2/3 power.

So you use the power rule, a special case of the chain rule:

Let u = 8 + 3x

f1(x) = u^2/3,  where  u = 8 + 3x.

I think you can handle it from here.

NOTE: You have redundant bracketing - [(8+3x)].  Typo?  Did you mistype the expression?  

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Paul Klarreich

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All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

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I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

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