[ASK] Derivative of an Algebraic Fraction find f(0) + f'(0)

In summary, the conversation discusses solving for the value of f(0) + f'(0) when given a function f(x). The steps for solving this problem are shown and the correct answer is determined to be -\frac{25}{36}, which is not one of the given options. The individual asking the question confirms the correctness of the solution and suggests double-checking the question.
  • #1
Monoxdifly
MHB
284
0
If \(\displaystyle f(x)=\frac{3x^2-5}{x+6}\) then f(0) + f'(0) is ...
A. 2
B. 1
C. 0
D. -1
E. -2

What I did:
If \(\displaystyle f(x)=\frac{u}{v}\) then:
u =\(\displaystyle 3x^2-5\) → u' = 6x
v = x + 6 → v' = 1
f'(x) =\(\displaystyle \frac{u'v-uv'}{v^2}\)=\(\displaystyle \frac{6x(x+6)-(3x^2-5)(1)}{(x+6)^2}\)
f(0) + f'(0) = \(\displaystyle \frac{3(0^2)-5}{0+6}\) + \(\displaystyle \frac{6(0)(0+6)-(3(0^2)-5)(1)}{(0+6)^2}\) = \(\displaystyle \frac{3(0)-5}{6}\) + \(\displaystyle \frac{0(0+6)-(3(0)-5)}{6^2}\)= \(\displaystyle \frac{0-5}{6}\) + \(\displaystyle \frac{0-(0-5)}{36}\) = \(\displaystyle \frac{-5}{6}\) + \(\displaystyle \frac{0-(-5)}{36}\) = \(\displaystyle \frac{-30}{36}\) + \(\displaystyle \frac{0+5}{36}\) = \(\displaystyle \frac{-25}{36}\)
The answer isn't in any of the options. I did nothing wrong, right?
 
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  • #2
Monoxdifly said:
If \(\displaystyle f(x)=\frac{3x^2-5}{x+6}\) then f(0) + f'(0) is ...
A. 2
B. 1
C. 0
D. -1
E. -2

What I did:
If \(\displaystyle f(x)=\frac{u}{v}\) then:
u =\(\displaystyle 3x^2-5\) → u' = 6x
v = x + 6 → v' = 1
f'(x) =\(\displaystyle \frac{u'v-uv'}{v^2}\)=\(\displaystyle \frac{6x(x+6)-(3x^2-5)(1)}{(x+6)^2}\)
f(0) + f'(0) = \(\displaystyle \frac{3(0^2)-5}{0+6}\) + \(\displaystyle \frac{6(0)(0+6)-(3(0^2)-5)(1)}{(0+6)^2}\) = \(\displaystyle \frac{3(0)-5}{6}\) + \(\displaystyle \frac{0(0+6)-(3(0)-5)}{6^2}\)= \(\displaystyle \frac{0-5}{6}\) + \(\displaystyle \frac{0-(0-5)}{36}\) = \(\displaystyle \frac{-5}{6}\) + \(\displaystyle \frac{0-(-5)}{36}\) = \(\displaystyle \frac{-30}{36}\) + \(\displaystyle \frac{0+5}{36}\) = \(\displaystyle \frac{-25}{36}\)
The answer isn't in any of the options. I did nothing wrong, right?
Your calculation is correct, and the answer is not one of the listed options. Maybe you should check whether you read the question correctly.
 
  • #3
Yes, the correct answer is [tex]-\frac{25}{36}[/tex].
 
  • #4
OK, thanks for the clarifications...
 

What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is the slope of a tangent line to the function at that point.

How do you find the derivative of an algebraic fraction?

To find the derivative of an algebraic fraction, you can use the quotient rule, which states that the derivative of a fraction is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the square of the denominator.

What is f(0) and f'(0)?

f(0) represents the value of the function at x=0, while f'(0) represents the derivative of the function at x=0. In other words, f(0) is the y-value at the point x=0, and f'(0) is the slope of the tangent line at x=0.

Why is it important to find f(0) + f'(0)?

Finding f(0) + f'(0) allows you to determine the behavior of the function at x=0. It can help you identify whether the function has a local maximum or minimum at that point, or if it is increasing or decreasing. It is also useful in finding the equation of the tangent line at x=0.

Can you provide an example of finding f(0) + f'(0)?

For example, if the function f(x) = (2x+1)/(x-3), then f(0) = 1 and f'(0) = 5. Therefore, f(0) + f'(0) = 6. This means that at x=0, the function has a y-value of 6 and a slope of 5, which can be used to determine the equation of the tangent line at x=0.

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