MACM 316: Calculus Review
Definition of Derivative:
df
The derivative of a function f (x) at a point x = a, denoted by f 0 (a) or (a), is
dx
f (x) − f (a) f (a + h) − f (a)
f 0 (a) = lim OR f 0 (a) = lim .
x→a x−a h→0 h
Equation of Tangent Line:
The equation of the tangent line to the curve y = f (x) at the point (a, f (a)) is
y = f (a) + f 0 (a)(x − a).
Derivative Rules:
Let c be a real constant, and f (x) and g(x) functions whose derivatives, f 0 (x) and g 0 (x), exist.
d
Derivative of a constant: c=0
dx
d c
Power rule: x = cxc−1 , c 6= 0
dx
d df
Constant multiple: (cf ) = c
dx dx
d df dg
Sum rule: (f + g) = +
dx dx dx
d df dg
Difference Rule: (f − g) = −
dx dx dx
d df dg
Product Rule: (f · g) = ·g+f ·
dx dx dx
df dg
d
f ·g−f ·
Quotient Rule: = dx dx
dx g g2
Chain Rule:
Let F (x) = f (g(x)) be the composition of the two functions f and g. If the derivatives g 0 (x) and
f 0 (g(x)) both exist, then the derivative F 0 (x) also exists and is given by
F 0 (x) = f 0 (g(x)) · g 0 (x)
Rules for Exponents:
1
xa xb = xa+b x−a =
x√a
xa /xb = xa−b x1/a = a
x
a b ab
(x ) = x x y = (xy)a
a a
Rules for Logarithms:
The function log x is defined only for x > 0. By default, the base is assumed to be e, so that
log x = loge x = ln x is the natural logarithm. Its inverse function is ex which is also denoted
exp(x). Logarithms with other bases are identified by explicitly including the base, such as
log10 x, log2 x, etc.
log(xy) = log x + log y
x
exp(log(x)) = log(exp(x)) = x log = log x − log y
y
log(xa ) = a log x
WARNING: log(x + y) 6= log x + log y !
Some Useful Plots:
Exponential Function, ex = exp(x) Logarithm Function, log x = ln x
20 3
2
15
1
10 0
log(x)
ex
−1
5 −2
−3
0
−4
−5 −5
−3 −2 −1 0 1 2 3 −2 0 2 4 6 8 10
x x
Derivatives of Special Functions:
d x d x
Exponential: e = ex a = ax log a
dx dx
d 1 d 1
Logarithm: log x = loga x =
dx x dx x log a
d d 1
Trig functions: sin x = cos x arcsin x = √
dx dx 1 − x2
d d −1
cos x = − sin x arccos x = √
dx dx 1 − x2
d d 1
tan x = sec2 x arctan x =
dx dx 1 + x2
d
sec x = sec x tan x
dx
Three Key Theorems:
1. Mean Value Theorem (MVT, text p. 4):
If f (x) is a continuous function for x ∈ [a, b] and is
also differentiable on (a, b), then there exists a number
c ∈ (a, b) with
f (b) − f (a)
f 0 (c) = .
b−a
2. Intermediate Value Theorem (IVT, text p. 6):
If f (x) is a continuous function for x ∈ [a, b] and K is
any number between f (a) and f (b), then there exists a
number c ∈ (a, b) for which f (c) = K.
3. Taylor’s Theorem (text p. 8):
Suppose that f (x) and its first n + 1 derivatives are continuous functions on the interval [a, b],
and let xo ∈ [a, b] be some given point. Then for every x ∈ [a, b], we can write
f (x) = Pn (x) + Rn (x)
where the nth order Taylor polynomial is
f 00 (xo ) f (n) (xo )
Pn (x) = f (xo ) + f 0 (xo )(x − xo ) + (x − xo )2 + · · · + (x − xo )n
2! n!
n
X f (k) (xo )
= (x − xo )k ,
k=0
k!
the remainder (or truncation error) term is
f (n+1) (c)
Rn (x) = (x − xo )n+1 ,
(n + 1)!
and c is some point lying between xo and x.
