State the domain of the function and the domain of its derivative

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The discussion focuses on finding the derivatives of several functions and determining their domains. For the function f(x) = x + √x, the derivative is incorrectly stated as x^{3/2}, while the correct derivative involves applying the power rule to √x. The second function, f(x) = (3 + x) / (1 - 3x), has its derivative correctly calculated as 10 / (1 - 3x)^2. The third function requires the quotient rule, and participants emphasize the importance of correctly applying differentiation rules. The fourth function involves confusion over the correct interpretation of the square root, highlighting the need for clarity in notation and application of the chain rule.
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Find the derivative. State the domain of the function and the domain of its derivative.

f(x) = x + √x

f(x) = (3 + x) / 1-3x

Find F'(a)

f(x) = (x^2 + 1) / (x - 2)

f(x) = √3x + 1
 
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And your work so far...?
 


For the first one i got up to x + √x = x^1/2 = x^3/2

2nd one 1(1-3x) - (-3)(3 + x) all over (1-3x)^2 = 10 / (1 - 3x) ^2

3rd one i know its the quotient rule but not sure where to go with this one

For the fourth i tried the chain rule

√3x + 1 = 3x^1/2 + 1 = 1/2(3x)^1/2
 


bballj228 said:
For the first one i got up to x + √x = x^1/2 = x^3/2
? No, x+ \sqrt{x} is NOT equal to x^{1/2}= \sqrt{x} and NEITHER of those is equal to x^{1/2}. Did you mean that \sqrt{x}= x^{1/2}? And that the derivative is x^{3/2}? That last is not correct, either. Surely, you know what the derivative of x= x1 is 1? And what is the derivative of \sqrt{x}= x^{1/2}?

2nd one 1(1-3x) - (-3)(3 + x) all over (1-3x)^2 = 10 / (1 - 3x) ^2
Yes, that is correct!

3rd one i know its the quotient rule but not sure where to go with this one
USE the quotient rule of course! What is (x2+ 1)' ? What is (x- 2)'? Do it just like you did number 2.

For the fourth i tried the chain rule

√3x + 1 = 3x^1/2 + 1 = 1/2(3x)^1/2

Is that √(3)x+ 1, √(3x)+ 1, or √(3x+1)? In any case, none if those is equal to 3x^(1/2)+ 1.

The derivative of √(3) x+ 1 should be trivial. √(3x)+ 1 can be done as √(3)x^(1/2)+ 1, and √(3x+1) should be done using the chain rule: √(3x+1)= √u with u= 3x+1:
(√(3x+1))'= (du^(1/2)/du)(d(3x+1)/dx).
 


HallsofIvy said:
? No, x+ \sqrt{x} is NOT equal to x^{1/2}= \sqrt{x} and NEITHER of those is equal to x^{1/2}. Did you mean that \sqrt{x}= x^{1/2}? And that the derivative is x^{3/2}? That last is not correct, either. Surely, you know what the derivative of x= x1 is 1? And what is the derivative of \sqrt{x}= x^{1/2}?

No, I don't know actually.
 


Use the power rule for differentiation. What is the derivative of x^n? Just apply it to x^1/2.
 

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