THE DERIVATIVE
Review Notes on Differentiation for MATH. 1920 with Practice Exercises
The Leibnitz notation for the derivative of f(x) with respect to x is . It is commonly used
df(x)
dx
whenever we are asked to find the derivative, but do not have the function given as an equation. It
is essential whenever it is not clear which letter in a function is the independent variable. It was used
to introduce every differentiation rule in MATH 1910:
See pages 192-6,198,202,205, 223-224,228-234,240,241,243,244,247-51,268,270, etc.
Several different notations for the first derivative are shown on p. 162. Higher order derivatives and
their notations are discussed on p. 165-8.
Notation is very important because poor notation misleads others who try to follow your work, and it
frequently misleads you as well. Do not put equal signs between quantities which are not equal. The
symbol has no meaning because we do not know what to differentiate!
d
dx
Find and simplify each of the following derivatives.
1. HINT: The variable is “a”; treat all other letters as constants.
d
da 234
(axyabc)
π
−+−+
2. 3. 4.
2
2
d
db 234
(axyabc)
π
−+−+
d
dy 234
(axyabc)
π
−+−+
d
dx 234
(axyabc)
π
−+−+
Some derivatives can be found using different methods. Whenever you are tempted to use a
cumbersome method such as the product or quotient rule, study the problem to see if an easier
approach will work. [See note at bottom of p. 205.] No matter which method you use, you must
simplify the answer!
Find and simplify each of the following derivatives, using the specified methods. [Sec. 3.1]
5. Use the quotient rule first. Then divide the denominator into the
d
dx
2
x4x3
x
++++
numerator and rework the problem using the power rule. Show that your
answers are equivalent.
6. Use the product rule first. Then distribute the monomial and rework the problem
d
dt 3t(t2)++
using the power rule. Show that your answers are equivalent.
Many differentiation problems require the use of the chain rule. The chain rule is never used alone,
but is always used in combination with at least one other rule. Some situations requiring the chain
rule are quantities raised to powers, roots of quantities, trig functions of quantities, exponents which
are quantities, etc. Sometimes other methods can be employed, but the chain rule is usually easier
and shorter. The answer needs to be in factored form.
