Thanks Mao 
I have another question:
If you have the equation x = 5, how do you find d(5) / dx ?
Would the answer be 1, since you treat 5 as the function of x and you take the derivative with respect to x, or would it be 0, since there is no change in the value of 5?
The answer is undefined. X=5 is a vertical line. It's gradient would be +/- inifinity.
Nooooo I'm not asking for the gradient. The gradient would be dy/dx.
I'm asking for d(5) / dx.
If you look at it from an algebraic perspective, you have 5 = x, therefore d(5) / dx = 1. But I'm not sure if it works that way which is what I'm asking.
But thanks for trying anyway.
firstly, x=5 is not a function of x.
secondly,
}{dx})
is equivalent to
with
, so your notation is incorrect.
Thirdly,
only makes sense if
is a function of
, that is,
)
. In this case,
cannot be expressed as a function of
, its differential is undefined.
An alternative approach is if we were to take
 = m(x-5))
(a straight line going through (5,0), then take the limit as
(i.e. as the line becomes so steep it is totally vertical). The derivative is
) = \lim_{m\to\infty} \frac{df}{dx} = \lim_{m\to\infty} m = \infty)
Alternatively the limit
is also a valid interpretation. Since the two limits do not agree (and the fact that they're both infinities), the derivative is undefined.
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