Math 175
Notes and Learning Goals
Lesson 0-1
This lesson is a quick review of the derivatives and antiderivatives from Calculus I.
• The average rate of change of f (x) on the interval [a, b] is
∆f
f (b) − f (a)
=
∆x
b−a
Where ∆f is the change in f and ∆x is the change in x.
• The average rate of change is the slope of the secant line that passes though (a, f (a)) and (b, f (b)).
• The derivative, f 0 (x), gives the instantaneous rate of change of f (x) at the point x
• The instantaneous rate of change is the slope of the tangent line to f (x) at the point (x, f (x)).
• The following are different notations that represents the derivative:
df
d
[f (x)] =
= f 0 (x)
dx
dx
• Derivatives of a function (in symbolic form) can be found by the set of derivative rules.
Basic Derivatives
• Constant Function:
• Power Rule:
d
n
dx [x ]
d
dx [c]
• Natural Logarithm:
=0
= nxn−1
• Exponential Function:
• Sine Function:
d
x
dx [e ]
d
dx [ln(x)]
d
dx [sin(x)]
=
1
x
= cos(x)
d
dx [cos(x)]
= ex
• Cosine Function:
= cf 0 (x)
d f (x)
f 0 (x)g(x) − g 0 (x)f (x)
• Quotient:
=
dx g(x)
[g(x)]2
= − sin(x)
Derivative Rules
• Constant Multiple:
• Sum/Difference:
• Product:
d
dx [cf (x)]
d
dx [f (x)
d
dx [f (x)g(x)]
± g(x)] = f 0 (x) ± g 0 (x)
= f 0 (x)g(x) + g 0 (x)f (x)
• Chain:
d
dx [f (g(x))]
= f 0 (g(x))g 0 (x)
Implicit Differentiation
Assume that y = f (x)
• Power Rule + Chain:
d
n
dx [y ]
dy
= ny n−1 y 0 = ny n−1 dx
• Chain Rule:
1
d
dx [f (y)]
dy
= f 0 (y)y 0 = f 0 (y) dx
Antiderivative/Indefinite Integral
• The opposite of the derivative is called the antiderivative.
• The antiderivative of f (x) is any function F (x) such that
F 0 (x) = f (x)
• To check if an antiderivative is correct, take its derivative (using the derivative rules) and see if you get the
original function back.
• The indefinite integral gives the family of antiderivatives
Z
f (x)dx = F (x) + C
• Since the derivative of a constant is 0, the family of antiderivatives has an unknown constant (+C) that is
called the constant of integration.
• Doing each basic derivative rule backwards gives a basic antiderivative rule.
• There is NO product, quotient or chain rules for the antiderivative.
Basic Antiderivative Rules
• Constant Function:
• Power Rule:
R
R
xn dx =
• Exponential Function:
• Natural Logarithm:
R
• Sine Function:
1
n+1
n+1 x
R
+C
(n 6= −1)
ex dx = ex + C
x−1 dx =
R
1
x dx
R
sin(x)dx = − cos(x) + C
R
• Cosine Function: cos(x)dx = sin(x) + C
R
R
• Constant Multiple: cf (x)dx = c f (x)dx
mdx = mx + C
• Sum/Difference:
R
R
R
(f (x) ± g(x))dx = f (x)dx ± g(x)dx
= ln(|x|) + C
2
Additional Function Derivatives
• The following is a list of additional derivatives that can be used as a reference.
• You are not required to memorize these rules, but you should know where to look them up.
– General Exponential:
– General Logarithm:
d
x
dx [a ]
= ln(a)ax
d
dx [loga (x)]
=
1
ln(a)
·
1
x
d
2
dx [tan(x)] = sec (x)
d
Cotangent Function: dx
[cot(x)] = − csc2 (x)
d
[sec(x)] = tan(x) sec(x)
Secant Function: dx
d
Cosecant Function: dx
[csc(x)] = − cot(x) csc(x)
– Tangent Function:
–
–
–
1
1 − x2
1
d
– Inverse Cosine: dx
[cos−1 (x)] = − √
1 − x2
1
d
– Inverse Tangent: dx
[tan−1 (x)] =
1 + x2
– Inverse Sine:
−1
d
(x)]
dx [sin
=√
3