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I did the following:

Taking the derivative of $x^4$, I'll have $4x^3$. The slope of the tangente line at the point $x_0$ is $4x_0^3$.

Now to find the tangent line:

$$\begin{eqnarray*} {y}&=&{mx+b} \\ {[x_0^4]}&=&{[4x_0^3][x_0]+b} \\ {[x_0^4]-[4x_0^3][x_0]}&=&{b} \\ {-3x_0^4}&=&{b} \end{eqnarray*}$$

Then:

$$\begin{eqnarray*} {y}&=&{[4x^3_0]x-3x_0^4} \end{eqnarray*}$$

Is it correct?

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That should be correct since you took the derivative, then evaluated it at the point. So, going back to the definition of a limit, you should be correct.

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Here's a nice formula for computing the line tangent to $y = f(x)$ at the point $(x_0, f(x_0))$:

$y = f'(x_0)x + (f(x_0) - f'(x_0)x_0)$.

Ex: $f(x) = x^4$, so $f'(x) = 4x^3$. Thus:

$y = (4x_0^3)x + x_0^4 - (4x_0^3)x_0 = (4x_0^3)x - 3x_0^4$.

Of course, the most important thing here is to understand why the formula is correct, but I'll leave that as an exercise for the reader. :)

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