Formulasheetderivatives_2.pdf
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!
(ax ) = ax x !
(e ) = e
df f (x) = dx
a
x
!
f ! (x)
&
f (x! ) − f (x) x! − x f (x + h) − f (x) h→0 h
=
x! →x
=
#
v ! (uv) = uv ! + u! v !
x
!
(xx ) = xx (1 +
d (c) = 0 c dx x d d (k · f (x)) = k (f (x)) dx dx d d d (f (x) ± g (x)) = (f (x))± (g (x)) dx dx dx u
'! 1 1 √ # $ =√ 2 2 2 x+ x +a x + a2 $! # 2 $! 2 x =− x = 2x
dy dy dz = · dx dz dx g (x)
y = f (z)
z =
!
(f (g (x))) = f ! (g (x)) · g ! (x)
y = φ (t) ) dy dy φ! (t) dt =( )= ! dx dx ψ (t) dt (
x = ψ (t)
u!1 u2 u3 · · · un + u1 u!2 u3 · · · un + · · · + u1 u2 u3 · · · un−1 u!n ! u "! vu! − uv ! = v v2 $ # # $ # $ y = (1 + x) 1 + x2 1 + x4 · · · 1 + x2n
(u1 u2 · · · un )
x)
=
y! dy dx
% dy %% dx %x=0
1+
!
( x) = x ! ( x) = − x ! 2 ( x) = x ! ( x) = −cosec2 x ! ( x) = x x ! (cosec x) = −cosec x x # −1 $! 1 x =√ 1 − x2 # −1 $! −1 x =√ 1 − x2 # −1 $! 1 x = 1 + x2 # −1 $! −1 x = 1 + x2 # −1 $! 1 √ x = |x| x2 − 1 # $! −1 √ cosec−1 x = |x| x2 − 1 1 ! ( x) = x 1 ! ( a x) = x a
dy dx dy dx
x+y =
(xy)
( ) dy (xy) x +y dx 1−y (xy) x (xy) − 1
= =
f (x, y) = 0 dy =− dx ∂f df = ∂x dx
(
∂f /∂x ∂f /∂y
)
f (x, y) = (xy) − x − y dy dx
= =
) ∂f /∂x ∂f /∂y ( ) y (xy) − 1 − 0 − x (xy) − 0 − 1 −
(
y = f (x) · g (x) · h (x)
|y| |y|
1 dy y dx
# $# $ y = (1 + x) 1 + x2 1 + x3 % %% % |1 + x| · %1 + x2 %%1 + x3 % % % % % |1 + x| + %1 + x2 % + %1 + x3 %
= =
y!
x y (n−1)
nth
y (n)
2
1 2x 3x + + 1 + x 1 + x2 1 + x3
=
y = f (x)
y = d2 y 2 x−3 = 2 dx x3
g(x)
x x
nth y=( x· (
y=e
x· (
y = (
x)
= =
x)
(c1 u + c2 v) x)
x
( −
2
x + x
x·
(
x
+
x
∂ ! ( x) ∂x * base +,
+( = (
x·(
x)
x
x)
x
x)
x−1
x
(
(
· −
"
-
(ax )!
=
2
(n)
n .
=
n
(n)
= c1 u(n) + c2 v (n)
u(n) v + nu(n−1) v (1) n (n − 1) (n−2) !! + u v + · · · + uv (n) 2!
Cr u(n−r) v (r)
u(0) = u
k=0
v (0) = v y (20)
y = x2
x
"
-
(−
x)
x) ·
x + x
x x·
% % % f1 (x) f2 (x) f3 (x) % % % ∆ (x) = %% g1 (x) g2 (x) g3 (x) %% % h1 (x) h2 (x) h3 (x) % % ! % % f1 (x) f2! (x) f3! (x) % % % ∆! (x) = %% g1 (x) g2 (x) g3 (x) %% % h1 (x) h2 (x) h3 (x) % % % % f1 (x) f2 (x) f3 (x) % % ! % +%% g1 (x) g2! (x) g3! (x) %% % h1 (x) h2 (x) h3 (x) % % % % f1 (x) f2 (x) f3 (x) % % % +%% g1 (x) g2 (x) g3 (x) %% % h!1 (x) h!2 (x) h!3 (x) % % 3 % x x x % −1 0 f (x) = %% 6 % p p2 p3 3 d p∈R (f (x)) x=0 dx3 x
(uv)
=
x ∂ ! ( x) ∂x * base +, (xn )!
y
x)
)
nth
v (x)
x)
dy =e dx
y dy dx
u (x)
x
y! ,
dy dx
(
% % % % % %
x)
) y
=
) ( ) 1 1 −1 + + ··· x2 + x + 1 x2 + 3x + 3 ( ) 1 −1 + + ··· x2 + 7x + 13 dy dx # √ $ √ √ −1 y= x 1 − x − x 1 − x2 dy dx −1
(
(ax ) = ax x !
(e ) = e
df f (x) = dx
a
x
!
f ! (x)
&
f (x! ) − f (x) x! − x f (x + h) − f (x) h→0 h
=
x! →x
=
#
v ! (uv) = uv ! + u! v !
x
!
(xx ) = xx (1 +
d (c) = 0 c dx x d d (k · f (x)) = k (f (x)) dx dx d d d (f (x) ± g (x)) = (f (x))± (g (x)) dx dx dx u
'! 1 1 √ # $ =√ 2 2 2 x+ x +a x + a2 $! # 2 $! 2 x =− x = 2x
dy dy dz = · dx dz dx g (x)
y = f (z)
z =
!
(f (g (x))) = f ! (g (x)) · g ! (x)
y = φ (t) ) dy dy φ! (t) dt =( )= ! dx dx ψ (t) dt (
x = ψ (t)
u!1 u2 u3 · · · un + u1 u!2 u3 · · · un + · · · + u1 u2 u3 · · · un−1 u!n ! u "! vu! − uv ! = v v2 $ # # $ # $ y = (1 + x) 1 + x2 1 + x4 · · · 1 + x2n
(u1 u2 · · · un )
x)
=
y! dy dx
% dy %% dx %x=0
1+
!
( x) = x ! ( x) = − x ! 2 ( x) = x ! ( x) = −cosec2 x ! ( x) = x x ! (cosec x) = −cosec x x # −1 $! 1 x =√ 1 − x2 # −1 $! −1 x =√ 1 − x2 # −1 $! 1 x = 1 + x2 # −1 $! −1 x = 1 + x2 # −1 $! 1 √ x = |x| x2 − 1 # $! −1 √ cosec−1 x = |x| x2 − 1 1 ! ( x) = x 1 ! ( a x) = x a
dy dx dy dx
x+y =
(xy)
( ) dy (xy) x +y dx 1−y (xy) x (xy) − 1
= =
f (x, y) = 0 dy =− dx ∂f df = ∂x dx
(
∂f /∂x ∂f /∂y
)
f (x, y) = (xy) − x − y dy dx
= =
) ∂f /∂x ∂f /∂y ( ) y (xy) − 1 − 0 − x (xy) − 0 − 1 −
(
y = f (x) · g (x) · h (x)
|y| |y|
1 dy y dx
# $# $ y = (1 + x) 1 + x2 1 + x3 % %% % |1 + x| · %1 + x2 %%1 + x3 % % % % % |1 + x| + %1 + x2 % + %1 + x3 %
= =
y!
x y (n−1)
nth
y (n)
2
1 2x 3x + + 1 + x 1 + x2 1 + x3
=
y = f (x)
y = d2 y 2 x−3 = 2 dx x3
g(x)
x x
nth y=( x· (
y=e
x· (
y = (
x)
= =
x)
(c1 u + c2 v) x)
x
( −
2
x + x
x·
(
x
+
x
∂ ! ( x) ∂x * base +,
+( = (
x·(
x)
x
x)
x
x)
x−1
x
(
(
· −
"
-
(ax )!
=
2
(n)
n .
=
n
(n)
= c1 u(n) + c2 v (n)
u(n) v + nu(n−1) v (1) n (n − 1) (n−2) !! + u v + · · · + uv (n) 2!
Cr u(n−r) v (r)
u(0) = u
k=0
v (0) = v y (20)
y = x2
x
"
-
(−
x)
x) ·
x + x
x x·
% % % f1 (x) f2 (x) f3 (x) % % % ∆ (x) = %% g1 (x) g2 (x) g3 (x) %% % h1 (x) h2 (x) h3 (x) % % ! % % f1 (x) f2! (x) f3! (x) % % % ∆! (x) = %% g1 (x) g2 (x) g3 (x) %% % h1 (x) h2 (x) h3 (x) % % % % f1 (x) f2 (x) f3 (x) % % ! % +%% g1 (x) g2! (x) g3! (x) %% % h1 (x) h2 (x) h3 (x) % % % % f1 (x) f2 (x) f3 (x) % % % +%% g1 (x) g2 (x) g3 (x) %% % h!1 (x) h!2 (x) h!3 (x) % % 3 % x x x % −1 0 f (x) = %% 6 % p p2 p3 3 d p∈R (f (x)) x=0 dx3 x
(uv)
=
x ∂ ! ( x) ∂x * base +, (xn )!
y
x)
)
nth
v (x)
x)
dy =e dx
y dy dx
u (x)
x
y! ,
dy dx
(
% % % % % %
x)
) y
=
) ( ) 1 1 −1 + + ··· x2 + x + 1 x2 + 3x + 3 ( ) 1 −1 + + ··· x2 + 7x + 13 dy dx # √ $ √ √ −1 y= x 1 − x − x 1 − x2 dy dx −1
(
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