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Formula Sheet - Calculus
1. Derivative Rules
(a) [f (x)g(x)]0 = f 0 (x)g(x) + f (x)g 0 (x) (product rule)
0
f (x)
f 0 (x)g(x) − f (x)g 0 (x)
(b)
=
(quotient rule)
g(x)
[g(x)]2
(c) [f (g(x))]0 = f 0 (g(x))g 0 (x) (chain rule)
2. Common Deivatives
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)
1
(A) = 0 where A is any constant
dx
1
(x) = 1
dx
1 n
(x ) = nxn−1
dx
1 x
(a ) = ax ln a
dx
1 x
(e ) = ex
dx
1
1
(ln x) = , x > 0
dx
x
1
1
(ln |x|) = , x 6= 0
dx
x
1
(sin x) = cos x
dx
1
(cos x) = − sin x
dx
1
(tan x) = sec2 x
dx
1
(cot x) = − csc2 x
dx
1
(sec x) = sec x tan x
dx
1
(csc x) = − csc x cot x
dx
1
1
(sin−1 x) = √
dx
1 − x2
1
1
(cos−1 x) = − √
dx
1 − x2
1
1
(tan−1 x) =
dx
1 + x2
1
3. Integration Rules
Z
Z
(a)
u dv = uv − v du (integration by parts)
(b) Know ‘integration by substitution’
(c) Know ‘partial fraction decomposition’ and that is IMPORTANT!
4. Common Integrations
Z
(a)
A dx = Ax + C where A is any constant
Z
Z
(b)
Af (x) dx = A f (x) dx where A is any constant
Z
(c)
ln x dx = x ln x − x + C
Z
1
(d)
eax dx = eax + C
a
Z
xn+1
(e)
xn dx =
+ C if n 6= −1
n+1
Z
1
(f)
dx = ln |x| + C
x
Z
ln |ax + b|
1
dx =
+C
(g)
ax + b
a
Z
(h)
sin x dx = − cos x + C
Z
(i)
cos x dx = sin x + C
Z
(j)
sec2 x dx = tan x + C
Z
(k)
csc2 x dx = − cot x + C
Z
(l)
sec x tan x dx = sec x + C
Z
(m)
csc x cot x dx = − csc x + C
Z
(n)
sec x dx = ln | sec x + tan x| + C
Z
(o)
csc x dx = − ln | csc x + cot x| + C
Z
1
1
−1 x
+C
(p)
dx
=
tan
a2 + x 2
a
a
Z
1
1 −1 x √
(q)
dx = sin
+C
a
a
a2 − x 2
2