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G. M. Fichtenholz – Integral and differential calculus, vol. 1, PWN Warsaw 1999 (originally in Russian)
Constant functions.
If y = const then always Δy = 0. It does not depend on the Δx value.
Polynomial functions.
If y = x^{n} where n is a natural number. Let's define Δx then Δy is
thus
and
when Δx → 0
1/x Function
If y = 1/x then y + Δy is
which gives
and finally when Δx → 0
x^{1/2} Function
If y = x^{1/2} (for x > 0) then
then
and finally
when Δx → 0
Power functions.
If y = x^{μ} where μ is a real number. Let's assume x ≠ 0
when Δx → 0
because when Δx/x → 0
Exponential functions.
If y = a^{x} where a > 0, -∞ < x < +∞
when Δx → 0
when a = e
Logarithmic functions.
If y = log_{a} x where 0 < a ≠ 1, 0 < x < +∞
when Δx → 0
In the case of natural logarithm
Trigonometric functions.
Let's consider y = sin(x) then
when Δx → 0
because when Δx → 0
Inverse functions.
Let's consider y = f(x) that has its inverse function and has in the point x_{0} finite and different from 0 derivative f'(x_{0}). Then also the derivative of its inverse function g(y) such that x = g(y) equals 1/f(x_{0}).
when Δx → 0
Cyclometric functions.
Let's consider y = arcsin(x) (-1 < x < 1 and -π/2 < y < π/2). y is the inverse function of x = sin(y). The derivative of x equals
it implies that also the derivative of y exists
the sign is set as plus because cos(y) > 0.
Theorem.
Let's assume that:
or shorter
y = const | y' = 0 |
y = x; | y' = 1 |
y = x^{μ}; | y' = μ x^{μ - 1} |
y = a^{x}; | y' = a^{x}ln a |
y = log^{a}(x); | y' = log^{a}(e)/x |
y = sin(x); | y' = cos(x) |
y = cos(x); | y' = -sin(x) |
y = tg(x); | y' = 1/cos^{2}(x) |
y = ctg(x); | y' = -1/sin^{2}(x) |
y = arcsin(x); | y' = 1/[1 - x^{2}]^{1/2} |
y = arccos(x); | y' = -1/[1 - x^{2}]^{1/2} |
y = arctg(x); | y' = 1/[1 + x^{2}] |
Let a function u = f(x) has in the point x the derivative u'.
then the function z = const u has also its derivative in the point x
Let a function v = g(x) has in the point x the derivative v'. Then the function y = u ± v has also the derivative in that point that equals
and
finally in the limit when Δx → 0
Let a function v = g(x) has in the point x the derivative v'. Then the function y = uv has also the derivative in that point that equals
and
finally in the limit when Δx → 0 also Δv → 0
Let a function v = g(x) differ from 0 and has in the point x the derivative v'. Then the function y = u/v has also the derivative in that point that equals
and
finally in the limit when Δx → 0 also Δv → 0
Presented-below examples only show the way of finding and presenting solutions.