WTW 165 Mathematics
Unit 2.1 Differentiation formulas and rules
Lecture notes
Dr. Valisoa Rakotonarivo
Objectives
1. Use the seven formulas to find the derivative function.
2. Use the five rules to find the derivative function.
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Remark
To find the formula of the derivative of some functions, we can use the
limit definition. In this course we will use the formulas without proofs,
but below is one example of using the limit definition to find the
derivative of a function.
2
Example (limit definition)
Use the limit definition to find the derivative of f (x) = x1 .
Solution:
1 1
f (x + h) − f (x) x+h − x
f ′ (x) = lim = lim
h→0 h h→0 h
−1
= lim
h→0 x(x + h)
−1
= 2.
x
3
Differentiation formulas
d
1. dx k = 0, k ∈ R.
d n n−1
2. dx x = n x for every real number n.
d x x
3. dx e = e .
d x x
4. dx a = (ln a)a , a > 0.
d 1
5. dx ln x = x , x > 0.
d
6. dx sin x = cos x
d
7. dx cos x = − sin x.
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Rules
1. Derivative of a constant multiple
d d
[kf (x)] = k f (x)
dx dx
2. Sum rule
d d d
[f (x) + g (x)] = f (x) + g (x)
dx dx dx
3. Derivative of a difference
d d d
[f (x) − g (x)] = f (x) − g (x)
dx dx dx
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Rules
4. Product rule
d d d
[f (x)g (x)] = f (x) g (x) + f (x) g (x)
dx dx dx
5. Quotient rule
d d
dx f (x) g (x) − f (x) dx g (x)
d f (x)
=
dx g (x) g 2 (x)
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Example 1
√ 1
1. Find the derivative of f (x) = x+ x2 + x 3 + 7.
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Example 1
2. Find g ′ (x) if g (x) = x 5 + 5x + 5e x + 5 ln(x).
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Example 1
3. Find f ′ (π) if f (z) = 8 cos z − 2 sin z.
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Example 1
4. Find the derivative of f (x) = x 3 cos x.
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Example 1
2−e x
5. Find f ′ (x) if f (x) = 3x+sin x .
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Example 1
6. Find the slope of the tangent line to the graph of f (z) = z 4 at the
point where z = −2.
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Example 1
7. Find all the x-values such that the tangent line to the polynomial
p(x) = 9x + 3x 2 − x 3 at x is parallel to the x-axis. The graph of the
polynomial p(x) = 9x + 3x 2 − x 3 is sketched below, so that you can
check your answer.
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Theorem (Revision)
Let I be an interval and let y = f (x) a function that is continuous and
differentiable on I :
• f ′ (x) > 0 on I ⇐⇒ f is increasing on I .
• f ′ (x) < 0 on I ⇐⇒ f is decreasing on I .
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Method: number line
When using the previous theorem, you have to use a number line. You
have to
1. Find the domain of the function y = f (x).
2. Use the formulas and rules and find f ′ (x). Because you will use the
formula, simplify your answer. Factor if possible.
3. Draw the number line (indicating the domain on the line).
4. Find the zero’s of the function y = f ′ (x) and indicate it on the
number line. The zero’s divided the number line into intervals.
5. Find the sign of the derivative function on each of these intervals by
substituting one value in the interval in the formula of y = f ′ (x) and
write down f ′ pos/neg on that interval.
6. Answer with a sentence ” The function in increasing on...”
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Example 2
Use a number line to find the interval(s) on which f (x) = x 4 − 2x 3 + 1 is
increasing and the interval(s) on which f is decreasing.
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Example 3
2t
Find the interval(s) on which f (t) = 4−t 2 is increasing.
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