Question

How do I find the Derivative of a Function where the x is a number?

If I was to find d/dx g(1) for example, would I first find the value of the function g(x) at x=1 and then find the derivative of that which becomes 0 because the result would probably be a constant. Or do I first find the derivative of g(x) and then plug in x=1 to calculate it?

Thanks

Answer 1

Given a function g(x), its derivative, if it exists, is equal to the limit

`g'(x) = \displaystyle\lim_(h\to0)\frac{g(x+h)-g(x)}h`

The limit is some expression that is itself a function of x. Then the derivative of g(x) at x = 1 is obtained by just plugging x = 1. In other words, find g'(x) - and this can be done with or without taking a limit - then evaluate g' (1).

Alternatively, you can directly find the derivative at a point by computing the limit

`g'(1) = \displaystyle\lim_(h\to0)\frac{g(1+h)-g(1)}h`

But this is essentially the same as the first method, we're just replacing x with 1.

Yet another way is to compute the limit

`g'(1) = \displaystyle\lim_(x\to1)(g(x)-g(1))/(x-1)`

but this is really the same limit with h = x - 1.

You do not compute g (1) first, because as you say, that's just a constant, so its derivative is zero. But you're not concerned with the derivative of some number, you care about the derivative of a function that depends on a variable.