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Summary - Mathematics

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Course
Mathematics

Institution
12th Grade

This summary sheet provides detailed working out on how to solve derivative problems. It specifically goes through the quotient rule, chain rule, product rule and how log functions are seen in derivatives.

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Uploaded on July 23, 2024
Number of pages 2
Written in 2023/2024
Type Summary

Institution Secondary school
Education 12th Grade
Course Mathematics
School year 3

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Derivatives Study Note

Solving Exponential Functions: Trig Rule:
- Apply chain rule 𝑓(𝑥) = 𝑥𝑠𝑖𝑛𝑥
- Look at 2 separate parts -product rule 𝑓(𝑥) = 𝑥 · 𝑠𝑖𝑛(𝑥)
2𝑥
Example: 𝑓(𝑥) = 𝑒 𝑓(𝑥) = 𝑠𝑖𝑛(2𝑥)
' '
Chain: ℎ (𝑔(𝑥)) · 𝑔 (𝑥) - Use chain rule - When the x is being
'
Where ℎ (𝑔(𝑥)) = original function multiplied
' Where g(x) = 2x and h(x) = sin x
- Hence multiply the function by 𝑔 (𝑥)
𝑥 𝑓(𝑥) = 3𝑠𝑖𝑛(𝑥)
𝑔(𝑥) = 2𝑥 ℎ (𝑥) = 𝑒
' ' 𝑥 - Just solve using basic trig rules
𝑔 (𝑥) = 2 ℎ (𝑥) = 𝑒 '
' 2𝑥 𝑓 (𝑥) = 3𝑐𝑜𝑠(𝑥)
𝑓 (𝑥) = 𝑒 · 2
Product Rule:
𝑥
- When 𝑒 is × # (5), The derivative - 2 different functions of x multiplied
is the same as the function together
𝑥 −𝑒
𝑥
Example: 𝑥𝑙𝑛𝑥 or
Ex: 𝑓(𝑥) = 5𝑒 or 𝑓(𝑥) = 2
𝑥 ' ' '
'
𝑓 (𝑥) = 5𝑒
𝑥
𝑓'(𝑥) =
−𝑒 RULE: 𝑓 (𝑥) = 𝑔 (𝑥) · ℎ(𝑥) + ℎ (𝑥) · 𝑔(𝑥)
2
- only 1 function so no product rule in place 1. Identify 2 functions multiplying
2. Identify g(x) and h(x) then take the
𝑥
RULE : 𝑎 = 𝑁 is equal to 𝑙𝑜𝑔 𝑎𝑁 = 𝑋 derivatives of each
3. Plug values into formula above
- Log base e = ln
Simplify
Log functions:
Rules: Chain Rule:
𝑙𝑛 3 When to use:
1. 𝑓(𝑥) = 𝑒 e and ln cancel out
- Have a function inside another other
𝑓(𝑥) = 3
function (inside always has x)
2
2. 𝑓(𝑥) = 2𝑙𝑛5 = 𝑓(𝑥) = 𝑙𝑛 5 - Raise to a power
6
3. 𝑥 = 𝑙𝑛8/3 𝑥 = 𝑙𝑛 8
3
Examples: (3𝑥 + 5) or 𝑙𝑛3𝑥 or (7𝑥 + 9
' '
Chain Rule: ℎ (𝑔(𝑥)) · 𝑔 (𝑥) 2
𝑐𝑜𝑠 (4𝑥) = 𝑐𝑜𝑠(4𝑥)
2

Example: 𝑙𝑛(2𝑥 + 1) ' '
ℎ (𝑔(𝑥)) · 𝑔 (𝑥)
1. First identify h(x) and g(x)
= 2(𝑐𝑜𝑠(4𝑥)) · (𝑑𝑦/𝑑𝑥 𝑐𝑜𝑠 4𝑥 )
𝑔(𝑥) = 2𝑥 + 1 ℎ (𝑥) = 𝑙𝑛 (𝑥)
= 2(𝑐𝑜𝑠(4𝑥)) · − 4𝑠𝑖𝑛4𝑥
' ' 1
𝑔 (𝑥) = 2 ℎ (𝑥) = 𝑥
' ' '
RULE: 𝑓 (𝑥) = ℎ (𝑔(𝑥)) · 𝑔 (𝑥)
'
2. Substitute 𝑔(𝑥) into ℎ (𝑥) to solve for g(x) = inner function
'
ℎ (𝑔(𝑥)) h(x) = outer function
1 *derivative of 𝑓(𝑥) ℎ(𝑔(𝑥)) is
= 2𝑥 +1 ' '
' ℎ (𝑔(𝑥) · 𝑔 (𝑥)
3. Multiply by 𝑔 (𝑥)

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