Calculus: Derivation rules
1. Derivative of a constant
The function f is constant . That is f(x) = b (b ∊ R)
so
lim (f(x + Δ x)  f(x))/Δx = (b  b)/Δx = 0
Δ x → 0
Rule 1:
f(x) constant
f'(x) = df(x)/dx = 0
2. Derivative of a power
The function f has the form:
f(x) = x^{n} (n )
f'(x) = dx^{n}/dx = lim (f(x + Δx)  f(x))/Δx
Δ x → 0
= lim ((x + Δx)^{n}  x^{n})/Δx
Δ x → 0
0/0 : indetermined.
We have:
a^{n}  b^{n} =
(a  b)(a^{n1} + a^{n2}b +
a^{n3}b^{2} + … + b^{n  1}
)
Therefore:
f'(x) = lim ((x + Δx)^{n}  x^{n})/Δx
Δ x → 0
= lim ((x + Δx  x)((x+δx)^{n1} + ...
+ x^{n1})/Δx
Δ x → 0
= lim ((x + δx)^{n1} + ... + x^{n1})
Δ x → 0
(x^{n1} + ... + x^{n1})
= n x^{n1}
Rule 2:
f(x) = x^{n}
f'(x) = n x^{n1}
3. Derivative of product of a
constant by a function
The function f has the form:
f(x) = k g(x)
k .
f'(x) = lim (k g(x + Δx)  k g(x))/Δx
Δ x → 0
= k lim ( g(x + Δx)  g(x))/Δx
Δ x → 0
= k g'(x)
Rule 3:
f(x) = k g(x)
f'(x) = k g'(x)
k is a constant
4. Derivative of the sum of two
functions
The function f has the form:
f(x) = g(x) ± h(x)
f'(x) = lim (g(x + Δx) ± h(x + Δx)  g(x)  h(x))/Δx
Δ x → 0
= lim (g(x + Δx)  g(x))/Δx
Δ x → 0
± lim (h(x + Δx)  h(x))/Δx
Δ x → 0
= g'(x) ± h'(x)
Rule 4:
f(x) = g(x) ± h(x)
f'(x) = g'(x) ± h'(x)
The rule is generalized to more
that two functions:
[f(x) + g(x) + h(x) + ... + m(x)]' = f'(x) + g'(x) + h'(x) + ... + m'(x)
5. Derivative of the product of two
functions
The function f has the form:
f(x) = g(x). h(x)
f'(x) = lim (g(x + Δx).h(x + Δx)  g(x).h(x))/Δx
Δ x → 0
= lim [g(x + Δx).h(x + Δx)  g(x).h(x)
+ g(x)h(x + Δx)  g(x)h(x + Δx)
]/Δx
Δ x → 0
= lim [(g(x + Δx)  g(x))h(x + Δx)
+ g(x)(h(x + Δx)  h(x))
]/Δx
Δ x → 0
= lim [g'(x)h(x + Δx)]
+ g(x)h'(x)
Δ x → 0
= g'(x)h(x) + g(x)h'(x)
Rule 5:
f(x) = g(x).h(x)
f'(x) = g'(x)h(x) + g(x)h'(x)
6. Derivative of the division of two
functions
The function f has the form:
f(x) = g(x)/h(x)
so
g(x) = f(x) h(x)
g'(x) = lim (g(x + Δx)h(x + Δx)  g(x)h(x))/Δx
Δ x → 0
Using the rule for the product of two functions, we
have:
g'(x) = f'(x)h(x) + f(x)h'(x)
Therefore
g'(x) = f'(x)h(x) + f(x)h'(x)
[g'(x)  f(x)h'(x)]/h(x) = f'(x)
Since
g(x)/h(x) = f(x)
We have:
f'(x)= [h(x) g'(x)  g(x)h'(x)]/(h(x))^{2}
Rule 6:
f(x) = g(x)/h(x)
f'(x)= [h(x) g'(x)  g(x)h'(x)]/(h(x))^{2}
7. Derivative: Chain Rule
There is an usefull method to find a derivative of
composite functions:
y = f(x) = g(h(x))
Let:
y = g(u) and
h(x) = u
We have:
Δy = g(h(x + Δx ))  g(h(x)), and
Δu = h(x + Δx)  h(x)
so
dy/dx =
lim Δy/Δx
Δx → 0
Since
Δy/Δx = (Δy/Δu).
(Δy/Δx), we get:
dy/dx =

lim (Δy/Δu) .
Δ x → 0
 lim (Δu/Δx)
Δ x → 0


When Δx → 0
Δu = h(x + Δx)  h(x) → 0,
so:
dy/dx = 
lim (Δy/Δu) .
Δ u → 0
 lim (Δu/Δx)
Δ x → 0


= (dy/du) . (du/dx)
Rule 7: Chain Rule
f(x) = g(h(x))
y = g(u)
u = h(x)
f'(x)= (dy/du) . (du/dx)
8. Derivative of a power function
y = f(x) = (g(x))^{n},
Let:
y = u^{n} and
g(x) = u
We have
dy/dx = (dy/du) . (du/dx)
= (nu^{n1}) . (du/dx)
=
(ng(x)^{n1}) . (dg(x)/dx)
Rule 8:
f(x) = (g(x))^{n}
u = g(x)
f'(x) = dy/dx = n(g(x))^{n1}. (dg(x)/dx)
9. Exercices:
