A: TABLE OF BASIC DERIVATIVES
Let u = u(x) be a differentiable function of the independent variable x , that is u ′ (x) exists.
(A) The Power Rule :
d {u n } = nu n−1 . u ′
dx
d { u } = 1 . u′
dx
2 u
Examples :
d {(x 3 + 4x + 1) 3/4 } = 3 (x 3 + 4x + 1) −1/4 . (3x 2 + 4)
4
dx
1
d { 2 − 4x 2 + 7x 5 } =
(−8x + 35x 4 )
2
5
dx
2 2 − 4x + 7x
d {c} = 0 , c is a constant
dx
d {π 6 } = 0 , since π ≅ 3. 14 is a constant.
dx
(B) The Six Trigonometric Rules :
d {sin(u)} = cos(u). u ′
dx
d {cos(u)} = − sin(u). u ′
dx
d {tan(u)} = sec 2 (u). u ′
dx
d {cot(u)} = − csc 2 (u). u ′
dx
d {sec(u)} = sec(u) tan(u). u ′
dx
d {csc(u)} = − csc(u) cot(u). u ′
dx
Examples :
d {sin(x 3 )} = cos(x 3 ). 3x 2
dx
d {cos x )} = − sin( x ). 1
dx
2 x
d {tan{ 5 )} = sec 2 (5x −2 ). (−10x −3 )
dx
x2
d [cot{sin(2x)}] = − csc 2 {sin(2x)}. 2 cos(2x).
dx
d {sec( 4 x )} = sec( 4 x ) tan( 4 x ). 1 x −3/4
4
dx
d {csc(8x − 7)} = − csc(8x − 7) cot(8x − 7). 8
dx
(C) The Six Hyperbolic Rules :
d {sinh(u)} = cosh(u). u ′
dx
d {cosh(u)} = sinh(u). u ′
dx
d {tanh(u)} =sech 2 (u). u ′
dx
d {coth(u)} = −csch 2 (u). u ′
dx
d {sech(u)} = −sech(u) tanh(u). u ′
dx
d {csch(u)} = −csch(u) coth(u). u ′
dx
Examples :
d {sinh( 3 x )} = cosh( 3 x ). 1 x −2/3
3
dx
d {cosh(sec(x)} = sinh{sec(x)}. sec(x) tan(x)
dx
d [tanh{x 3 + sin(x 2 )}] =sech 2 {x 3 + sin(x 2 )}. (3x 2 + 2x cos(x 2 ))
dx
d {coth( 1 + 2x)} = −csch 2 ( 1 + 2x). (− 1 + 2)
x
x
dx
x2
d {sech{9x)} = −sech(9x) tanh(9x). 9
dx
d [csch{sinh(3x)}] = −csch{sinh(3x)} coth{sinh(3x)}. 3 cosh(3x)
dx
(D) The Exponential & Logarithmic Rule :
d {e u } = e u . u ′
dx
d {ln| u |} = u ′
u
dx
d {a u } = a u . ln(a). u ′
, a ∈ R, a > 0, a ≠ 1
dx
d {log | u |} = 1 u ′ ,
a
dx
ln(a) u
a ∈ R, a > 0, a ≠ 1
Examples :
d {e } = e −x 3 . (−3x 2 )
dx
d {ln| x 3 + 5x + 6 |} = 3x 2 + 5
dx
x 3 + 5x + 6
d {2 sec(x) } = 2 sec(x) . ln(2). sec(x) tan(x)
dx
2
d {log | tan(x) |} = 1 sec (x)
4
dx
ln(4) tan(x)
−x 3
(E) The Six Inverse Trigonometric Functions :
d {sin −1 (u)} =
u′
dx
1 − u2
d {cos −1 (u)} = −
u′
dx
1 − u2
d {tan −1 (u)} = u ′
dx
1 + u2
d {cot −1 (u)} = − u ′
dx
1 + u2
d {sec −1 (u)} =
u′
dx
|u| u 2 − 1
d {csc −1 (u)} = −
u′
dx
|u| u 2 − 1
(F) The Inverse Hyperbolic Functions :
d {sinh −1 (u)} =
u′
dx
1 + u2
d {cosh −1 (u)} =
dx
u′
u −1
2
d {tanh −1 (u)} = u ′
dx
1 − u2
Examples :
8x
d {sin −1 (4x 2 )} =
dx
1 − 16x 4
d {cos −1 (3x)} = −
3
dx
1 − 9x 2
1
2
x
d {tan −1 ( x )} =
1
=
1+x
dx
2 x (1 + x)
d {cot −1 (e x )} = − e x
dx
1 + e 2x
d [sec −1 (x 4 )} =
4x 3
4x 3
=
dx
|x 4 | x 8 − 1
x4 x8 − 1
d {csc −1 (2x)} = −
2
1
=−
2
dx
|2x| 4x − 1
|x| 4x 2 − 1
Examples :
1/x
1 + ln 2 (x)
d {sinh −1 (ln(x)} =
dx
5
25x 2 − 1
− 22
d {tanh −1 ( 2 )} =
x
= 2−2
x
4
dx
x −4
1− 2
x
d {cosh −1 (5x)} =
dx
(G) The Product and Quotient Rules :
d {uv} = u ′ v + uv ′
dx
d {ku} = ku ′ , k is a constant
dx
Examples :
d {x 3 ln(5x + 1)} = 3x 2 ln(5x + 1) + x 3 5
5x + 1
dx
3
2
d { x } = 1 d {x 3 } = 1 . 3x 2 = 3x
4 dx
4
4
dx 4
d { u } = u ′ v − uv ′
dx v
v2
d { tan(2x) } = 2 sec (2x) .x − tan(2x) .3x
dx
x3
x6
2
3
In table above it is assumed that u = u(x) and v = v(x) are differentiable functions
2
B: TABLE OF BASIC INTEGRALS
Let r , a , b , and β ∈ R , r ≠ −1 , a ≠ 0 , and β > 0.
(A) The Power Rule :
(ax + b) r+1
r
(ax
+
b)
dx
=
+C
∫
a(r + 1)
Examples :
∫ x −5 dx = − 14 x −4 + C , ∫(3x − 1) −2 dx =
∫ dx = ∫ 1 dx = x + C
∫ ax1 + b dx = 2a ax + b + C
∫ 7dx = 7 ∫ dx = 7x + C
∫ x 1+ 4 dx = 2 x + 4 + C.
(B) The Six Trigonometric Rules :
∫ sin( ax + b))dx = − 1a cos( ax + b) + C
∫ cos( ax + b)dx = 1a sin( ax + b) + C
∫ tan( ax + b)dx = 1a ln|sec( ax + b)|+C
∫ cot( ax + b)dx = 1a ln|sin( ax + b)|+C
∫ sec( ax + b)dx = 1a ln|sec( ax + b) + tan( ax + b)|+C
∫ csc( ax + b)dx = 1a ln|csc( ax + b) − cot( ax + b)|+C
(3x − 1) −1
+C
−3
Examples :
∫ sin(9x − 2)dx = − 19 cos(9x − 2) + C
∫ cos(3x)dx = 13 sin(3x) + C
∫ tan(5w − 1)dw = 15 ln|sec(5w − 1)|+C
∫ cot(1 − 7u)du = − 17 ln|sin(1 − 7u)|+C
∫ sec(3x)dx = 13 ln|sec(3x) + tan(3x)|+C
∫ csc(2t)dt = 12 ln|csc(2t) − cot(2t)|+C
(C) Additional Trigonometric Rules :
Examples
∫ sec 2 ( ax + b)dx = 1a tan( ax + b) + C
∫ sec 2 (2u/3)du = 32 tan(2u/3) + C
1 cot( w ) + C = −2 cot( w ) + C
∫ csc 2 ( ax + b)dx = − 1a cot( ax + b) + C
∫ csc 2 ( w2 )dw = − 1/2
2
2
1
1
∫ sec( ax + b) tan( ax + b)dx = a sec( ax + b) + C ∫ sec(3u) tan(3u)du = 3 sec(3u) + C
∫ csc( ax + b) cot( ax + b)dx = − 1a csc( ax + b) + C ∫ csc(5x) cot(5x)dx = − 15 csc(5x) + C
(D) The Six Hyperbolic Rules :
∫ sinh( ax + b)dx = 1a cosh( ax + b) + C
∫ cosh( ax + b)dx = 1a sinh( ax + b) + C
∫ tanh( ax + b)dx = 1a ln[cosh( ax + b)] + C
∫ coth( ax + b)dx = 1a ln|sinh( ax + b)|+C
∫ sech( ax + b)dx = 2a tan −1 (e ax+b ) + C
∫ csch( ax + b)dx = 1a ln|tanh( ax + b)/2|+C
Examples
∫ sinh(2x − 7)dx = 12 cosh(2x − 7) + C
∫ cosh( 2x5 )dx = 52 sinh( 2x5 ) + C
∫ tanh(2u)du = 12 ln[cosh(2u)] + C
∫ coth(x + 3)dx = ln|sinh(x + 3)|+C
∫ sech(3x − 6)dx = 23 tan −1 (e 3x−6 ) ++C
∫ csch(10t)dt = 101 ln|tanh(5t)|+C
(E) Additional Hyperbolic Rules :
Examples
∫ sech 2 ( ax + b)dx = 1a tanh( ax + b) + C
∫ csch 2 ( ax + b)dx = − 1a coth( ax + b) + C
∫ sech 2 (4w)dw = 14 tanh(4w) + C
∫ csch 2 (2u)du = − 12 coth(2u) + C
sech(3x)
∫ sech( ax + b) tanh( ax + b)dx = − 1a sech( ax + b) + C ∫ sech(3x) tanh(3x)dx = − 3 + C
∫ csch( ax + b) coth( ax + b)dx = − 1a csch( ax + b) + C ∫ csch( 3x ) coth( 3x )dx = −3csch(x/3) + C
(F) Exponential /Logarithmic Rules :
∫ e ax+b dx = 1a e ax+b + C
1 . k ax+b + C , 0 < k ∈ R , k ≠ 1.
∫ k αx+β dx = a ln(k)
∫ ax1+ b dx = 1a ln| ax + b|+C
Examples :
∫ e 7x dx = 17 e 7x + C
∫ 2 10x−17 dx = 101ln 2 2 10x−17 + C
∫ 2x 1− 3 dx = 12 ln|2x − 3|+C
.
(G) The Three Inverse Trigonometric Functions :
∫ β 21− x 2 dx = sin −1 ( βx ) + C
Examples :
∫ β 2 1+ x 2 dx = β1 tan −1 ( βx ) + C
∫ 3 +1 x 2 dx =
1 tan −1 ( x ) + C
3
3
∫ x x 12 − 4 dx = 12 sec −1 ( 2x ) + C , x > 2.
∫ x x 21− β 2 dx = β1 sec −1 ( βx ) + C , x > β
(H) The Three Inverse Hyperbolic Functions :
∫ β 21+ x 2 dx = sinh −1 ( βx ) + C
Examples :
1
dx = cosh −1 ( x ) + C
β
2
x −β
∫
1
dx = sin −1 (x/4) + C
2
16 − x
∫
2
∫ β 2 1− x 2 dx = β1 tanh −1 ( βx ) + C , |x|< β
∫
1
dx = sinh −1 (x) + C
2
1+x
∫
1
dx = cosh −1 (x/ 5 ) + C
x −5
2
∫ 36 1− x 2 dx = 16 tanh −1 ( 6x ) + C ,
(I) The Fundamental Theorems
b
|x|< 6
Examples :
e3
x=b
x=e
= g(b) − g(a)
= ln(e 3 ) − ln(e) = 3 − 1 = 2
∫ a f(x)dx = g(x)| x=a
∫ e 1x dx = ln|x| | x=e
d { v(x) F(t) dt = F(v(x)). v ′ (x) − F(u(x)). u ′ (x)
d { x cos(t 2 )dt = cos(x 4 ). 2x − cos(x 2 ). 1
∫
u(x)
dx
dx ∫ x
3
2
In table above it is assumed that :
(1) The function f(x) is continuous on [a, b] and ∫ f(x) dx = g(x) + C.
(2) The functions u(x) and v(x) are differentiable and ∫
v(x)
u(x)
F(t) dt exists.
C: BASIC TRIGONOMETRIC IDENTITIES
GROUP (A) :
(i) tan(θ) =
sin(θ)
cos(θ)
(ii) cot(θ) =
cos(θ)
sin(θ)
(iii) sec(θ) =
1
cos(θ)
(iv) csc(θ) =
1
sin(θ)
GROUP (B) :
(i) cos 2 (θ) + sin 2 (θ) = 1
(ii) 1 + tan 2 (θ) = sec 2 (θ)
(iii) cot 2 (θ) +1 = csc 2 (θ)
(ii) cos(2θ) = 2 cos 2 (θ) − 1
(iii) cos(2θ) = 1 − 2 sin 2 (θ)
GROUP (C) :
(i) sin(2θ) = 2 sin(θ) cos(θ)
(iv) cos 2 (θ) = 1 [1 + cos(2θ)]
2
(v) sin 2 (θ) = 1 [1 − cos(2θ)]
2
GROUP (D)
(i) sin(−θ) = − sin(θ)
(ii) cos(−θ) = cos(θ)
GROUP(E)
(i) cos(θ ± φ) = cos(θ) cos(φ) ∓ sin(θ) sin(φ)
(ii) sin(θ ± φ) = sin(θ) cos(φ) ± cos(θ) sin(φ)
GROUP (F)
(i) cos(θ) cos(φ) = 1 [cos(θ − φ) + cos(θ + φ)]
2
(ii) sin(θ) sin(φ) = 1 [cos(θ − φ) − cos(θ + φ)]
2
(iii) sin(θ) cos(φ) = 1 [sin(θ − φ) + sin(θ + φ)]
2
(iii) tan(−θ) = − tan(θ).
D: SPECIAL TRIGONOMETRIC EQUATIONS
(i) sin(x) = 0  x = nπ
(ii) cos(x) = 0
x =
(2n − 1)
π
2
where n is an integer : n = 0, ±1, ±2, ±3, . . .
E: HYPERBOLIC FUNCTIONS
(i) sinh(x) = 1 [e x − e −x ]
2
(iv) coth(x) =
cosh(x)
sinh(x)
(vii) cosh 2 (x) − sinh 2 (x) = 1
(iii) tanh(x) =
sinh(x)
cosh(x)
(vi) csch(x) =
1
sinh(x)
(ii) cosh(x) = 1 [e x + e −x ]
2
(v) sech(x) =
1
cosh(x)
(viii) 1 − tanh 2 (x) = sech 2 (x)
(ix) coth 2 (x) − 1 = csch 2 (x)
F:PROPERTIES OF LOGARITHMS
Let x and y be positive real numbers.
(i) ln(x) + ln(y) = ln(xy)
(ii) ln(x) − ln(y) = ln( xy )
(iv) ln(e k ) = k
(v) e ln(x) = x
(iii) ln(x m ) = m ln(x).
(vi) ln(1) = 0 , ln(e) = 1.
G:SPECIAL VALUES
(i) sin(0) = 0
(ii) cos(0) = 1
(iii) tan(0) = 0
(iv) sinh(0) = 0
(v) cosh(0) = 1
(vi) tanh(0) = 0
(vii) sin(nπ) = 0 and cos(nπ) = (−1) n , provided that " n " is an integer.
END