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Can someone help me solve this?. I have hard time understanding the lesson, and our teacher will give us a quiz next meeting. She leave this as our exercise. I just want to see the answer and solution. I can't comprehend of what she had been saying and of her given examples.

Find the derivative of  $$ f(x)= x^2+x+2 $$

Matti P.
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Suan Suan
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    What do you know about the derivative in general? Have you learned any of its definitions? – Andrew Chin Dec 11 '20 at 10:41
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    Do you know, for example, what the derivative of $x^2$ would be? It's one of the first ones that are taught on such courses ... – Matti P. Dec 11 '20 at 10:42
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    Usually, this question is followed by either a) using the definition of the derivative or b) using the theorems concerning the derivative, for example $\frac{d}{dx} x^2=2x$. If you haven't been shown these theorems, then you should answer by a) using the definition. But if you know the theorems, then apply them. – MasB Dec 11 '20 at 10:47
  • Is she asking you to find the derivative from first principles, as a limit? – Paul Dec 11 '20 at 10:51
  • No, but she had given an example solution using limits – Suan Suan Dec 11 '20 at 10:54
  • My teacher shows an example with solution, as a limit, but when I tried to copied what she had done. I cannot get the correct answer. – Suan Suan Dec 11 '20 at 11:19

2 Answers2

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Your teacher used limits, so I’m assuming that you’re required to use first principles.

The derivative of a function $f(x)$ is defined as $$f’(x)=\lim_{t\to x} \frac{f(t)-f(x)}{t-x} $$ Now just plug in the definition of $f$: $$f’(x) = \lim_{t\to x} \frac{(t^2+t+2)-(x^2+x+2)}{t-x} =\lim_{t\to x} \frac{(t-x)(t+x) +(t-x)}{t-x}=\lim_{t\to x} (t+x+1) = x+x+1 =2x+1$$

Vishu
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For $f(x) = x^n$ for $n \in \mathbb{R}$ then the derivative can be written

\begin{align} \frac{d}{dx} f(x) &= \frac{d}{dx}(x^n) \\ &=n x^{n-1} \end{align}

Constants, like $2$ can be written as $2x^0$, and the above can still be implemented.