Differentiable functions and examples

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can someone give me an example of Differentiable function at x=4 and funcstions who dont Differentiable function at x=4?

$f(x) = 2x-7$

$k(x) = 100x^7-55x^5+10000x^2$

$g(x) = 23$

Those are Differentiable function at x-4, right?

$q(x) = x/(x-4)$

$y(x) = 78x^2/(x^2-8x+16)$

$p(x) = 2/(x^2+16)$

and those are not Differentiable function?

Am I right?

Thanks for help

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1 Answer

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The first three examples that you provide are differentiable at $x=4$ and that's because they are polynomials and on $\mathbb{R}$ all polynomials are differentiable.

Also, you can check if a function is differentiable at $a$ if, simply, $f'(a)$ exists or if it has one of the following:

1. Vertical Tangent
2. Discontiunity
3. A corner like $\vee$ or $\wedge$

So, here the derivatives of your last three functions:

$$q'(x) = \frac{4}{(x-4)^2}$$ $$y'(x) = \frac{-624x}{(x-4)^3}$$ $$p'(x) = \frac{-4x}{(x^2+16)^2}$$

If you sub in $x=4$ here then you can find out which are not differentiable at $x=4$ and which are (hint: $p(x)$ but why and not simply because $f'(a)$ exists)

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