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if $f(3)=-2$ and $f'(3)=5$, find $g'(3)$ if,
$g(x)=3x^2-5f(x)$

the answer is -7, I find that very hard to understand the question. thanks

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What have you tried? –  user21436 Apr 22 '12 at 12:10
4  
$g(x) = 3x^2 - 5f(x)$ if you derive this expression wrt $x$, you get $g'(x) = 6x - 5f'(x)$ (assuming everything is nice and differentiable). Plugging the numbers yields the answer. –  tibL Apr 22 '12 at 12:13

4 Answers 4

up vote 2 down vote accepted

$g'(x)=6*x-5*f'(x)$

$g'(3)=6*3-5*f'(3)$=$18-25=-7$ i hope it would help you

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good lucks @Sb Sangpi –  dato datuashvili Apr 22 '12 at 12:35
    
sorry 4 another question,i did this another question $\frac{3x+1}{f(x)}$ it doesn't work. –  Sb Sangpi Apr 22 '12 at 13:43

Hint:

  • $(\alpha+\beta)\ '=\alpha\ '+\beta\ '$

  • $f$ is differentiable at $x=3$ as $f\ '(3)$ exists.

  • $f(3)$ is not a relevant piece of information here...

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sorry 4 another question,i did this another question $\frac{3x+1}{f(x)}$ it doesn't work. can u help? thx –  Sb Sangpi Apr 22 '12 at 13:46
    
@SbSangpi The key is quotient rule for taking derivatives--One hint is that you need the information about $f(3)$...Please try this out on your own. If you still have difficulties--ping me back here. Regards, –  user21436 Apr 23 '12 at 5:09

The solution to the problem comes from the fact that differentiation is a linear operator. This means that $(cf(x) + dg(x))' = cf'(x) + dg'(x)$ where $c$ and $d$ are constants. Assuming we know this and the differentiation rule for powers ($(x^n)' = n x^{n-1}$) we can continue by differentiating the equation $$g(x) = 3x^2 - 5f(x)$$ to get $$g'(x) = 6x - 5f'(x).$$

Therefore $g'(3) = 6\cdot 3 - 5 f'(3) = 18 - 5 \cdot 5 = 18 - 25 = -7$.

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Since it's a homework question here are some tips.

  1. Are $f,g$ differentiable functions everywhere??If not, how can you find the derivative of $g$ at a specific point (in our case 3).
  2. Find everything needed for calculating $g'(3)$ from the equation given.
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