244
III - Convergence: Continuous variables
§4. Differentiable functions
14 - Derivatives of a function
We have not yet properly introduced the concept of the derivative of a function of a real variable, except in a rather schematic way in Chap. II, nO 4,
though we used it in Chap. II a propos power series and the exponential and
trigonometric functions. It is time to examine it more closely.
First let us recall what a derivative is. Consider a function f with complex
values defined on an interval 24 I c lR not reducing to a single point. Given
an a E I, we examine the behaviour of f(a + h) as h tends to 0, implicitly
assuming, in all that follows, that a + h remains in I. If f is continuous at
the point a, then f(a + h) tends to f(a), in other words, the function f is
"almost constant" on a neighbourhood of a as we explained profusely in nO 2.
But instead of approximating f on a neighbourhood of a by the constant
function x ~ f(a), one might try to approximate it by a function a little less
simple, for example a linear function of the form g(x) = ex + d.
The least that one can ask is that it should be equal to f at the point a,
whence the condition ca + d = f(a). Then
(14.1)
g(x) = c(x - a) + f(a),
whence
(14.2)
g(a + h) = ch + f(a).
It remains to choose the constant c as well as possible.
Now the error committed in replacing f by 9 is given by
f(a + h) - g(a + h) = f(a + h) - f(a) - ch = h [f(a + h~ - f(a) - c] .
To minimise this we should choose c so that the coefficient of h is as small as
possible and, even better, tends to 0 with h. This means that we must choose
(14.3)
c = lim f(a + h~ - f(a).
h~O
h;'O
When this limit exists - and, of course, it does not always do so -, we say
that c is the derivative of f at a, denoted by f'(a); so the best way of approximating f on a neighbourhood of a by a linear function is to choose the
function
(14.4)
y = f'(a)(x - a) + f(a);
24 or more generally on an open subset X of JR, since an open set is an interval
on a neighbourhood of any of its points. For example, for the case of a rational
fraction f(x)/g(x), X is the set of x E R where g(x) l' O.
III - Convergence: Continuous variables
§4. Differentiable functions
14 - Derivatives of a function
We have not yet properly introduced the concept of the derivative of a function of a real variable, except in a rather schematic way in Chap. II, nO 4,
though we used it in Chap. II a propos power series and the exponential and
trigonometric functions. It is time to examine it more closely.
First let us recall what a derivative is. Consider a function f with complex
values defined on an interval 24 I c lR not reducing to a single point. Given
an a E I, we examine the behaviour of f(a + h) as h tends to 0, implicitly
assuming, in all that follows, that a + h remains in I. If f is continuous at
the point a, then f(a + h) tends to f(a), in other words, the function f is
"almost constant" on a neighbourhood of a as we explained profusely in nO 2.
But instead of approximating f on a neighbourhood of a by the constant
function x ~ f(a), one might try to approximate it by a function a little less
simple, for example a linear function of the form g(x) = ex + d.
The least that one can ask is that it should be equal to f at the point a,
whence the condition ca + d = f(a). Then
(14.1)
g(x) = c(x - a) + f(a),
whence
(14.2)
g(a + h) = ch + f(a).
It remains to choose the constant c as well as possible.
Now the error committed in replacing f by 9 is given by
f(a + h) - g(a + h) = f(a + h) - f(a) - ch = h [f(a + h~ - f(a) - c] .
To minimise this we should choose c so that the coefficient of h is as small as
possible and, even better, tends to 0 with h. This means that we must choose
(14.3)
c = lim f(a + h~ - f(a).
h~O
h;'O
When this limit exists - and, of course, it does not always do so -, we say
that c is the derivative of f at a, denoted by f'(a); so the best way of approximating f on a neighbourhood of a by a linear function is to choose the
function
(14.4)
y = f'(a)(x - a) + f(a);
24 or more generally on an open subset X of JR, since an open set is an interval
on a neighbourhood of any of its points. For example, for the case of a rational
fraction f(x)/g(x), X is the set of x E R where g(x) l' O.