Cheat Sheet: Techniques in Differentiation - 1 - Chain Rule

Oct 13, 2026

Cheat Sheet: Techniques in Differentiation - 2 - Chain Rule provides concise rules and worked examples for differentiating common function forms used in VCE calculus. It covers basic derivative rules, chain rule with “inside function” identification, quick chain rule shortcuts, exponential, logarithmic and trigonometric differentiation patterns, plus product rule and quotient rule. Includes step-by-step method guidance for rewriting expressions and evaluating derivatives at specific points.

LanguageEnglish

CategoryPresentations

FormatPDF

Pages2

Reading Time1 min

AccessFree to download

Basic Rules

Chain Rule

Quick Chain Rule

Definitions and Notation

Exponential and Logarithmic Rules

Trigonometric Functions

Product Rule

Specific Points

Quotient Rule

| Cheat Sheet: Techniques in Differentiation – 1 | | | | | | | --- | --- | --- | --- | --- | --- | | | | | | | | | Basic Rules | | Chain Rule | | | | | ddx~~ ~~ ( axn ) = anxn−1 , n ≠ 0
ddx~~ ~~ (x ) = 1
ddx~~ ~~ ( k ) = 0
ddx~~ ~~ ( ax−n ) =−anx− −n1 =− ~~ ~~xan1~~ ~~ , n ≠ 0 ddx~~ ~~ ( f (x) ) = f '(x)
ddx~~ ~~ (af (x) ) = a ~~ ~~ddx~~ ~~ ( f (x) ) = af '(x) ddx~~ ~~ ( f (x) ± g(x) ) = f '(x) ± g '(x)
ddx~~ ~~ ( sin x ) = cos xddx~~ ~~ ( cos x ) =− sin xddx~~ ~~ ( tan x ) = sec2 xddx~~ ~~ (ex ) = exddx~~ ~~ (log e x ) = ~~ ~~1x~~ ~~ , x > 0
DO NOT lower powers on trigonometric, logarithmic or exponential expressions. | | ddx~~ ~~ [ f (g(x)) ] = f '(g(x))g '(x)
Given y = f (u) where u is a function of x , then: dydx = dydu~~ ~~ × ~~ ~~ddux Note: u = the inside function
u = bracketed terms
u = power in exponentials
u = base numeral in logarithms u = angle in trigonometric functions | | | | | | | Example: ddx~~ ~~ (5(1− x2 )4 ) | | Example: ddx~~ ~~ (−3loge (tan x) ) | | | | | Let u = 1 − x 2
Let y = 5u4
dy dy du
= ×
dx du dx | du
∴ = −2x dx
∴ ~~ ~~dydu = 20u3 | Let u = tan x
Let y =−3log e u | du 2
∴ = sec x dx dy 3
∴ = − du u | | | | | | dy dy du ~~ ~~ = ~~ ~~ × ~~ ~~dx du dx
3 2 3sec2 x
= − . sec x = − u tan x | | | | | = 20u3 × −2x = −40xu3 =−40x(1− x2 )3 | | | | | | | | | | | | | | Quick Chain Rule | | | | | | | Brackets with high, fractional or negative powers:
ddx~~ ~~ a ( f (x) )n = anf '(x) ( f (x) )n−1
dydx = Power × Derivative of contents of bracket × Given expression but lower the power on the brackets by one
Example: ddx~~ ~~ (5(1− x2 )4 ) = 4× −2x ×5(1− x2 )3 = −40x(1− x2 )3 | | | | | | | | | | | | Definitions | | | | | | | | | | | | | | | | Quick Chain Rule | | | | | Derivative:
The gradient of a tangent to a curve
Notations:
y ' , ddx , dydx , d~~ ~~[fdx(x)], f '(x) , Dx [ y],
Dx [ f (x)], lim f (x + h) − f (x)
h→0 h | | | | | | | | | Exponential functions (base e only) ddx~~ ~~ ( aef ( x) ) = af '(x)ef ( x)
dydx = Derivative of the power × Given term | | |
ACE YOUR VCE | | | | d 􀂧 3esin x 􀂷 3esin x 3esin x . cos x
Example: dx 􀂩􀂨 − 2 􀂹􀂸 = cos x × − 2 = − 2 | | | | © [www.](www.tsfx.edu.au)[tsfx.edu.au](www.tsfx.edu.au) 2021 | VCE Lectures | A+ Notes & SACS | Tuition Classes | Trial Exams | Study Skills | Exam Strategy Cheat Sheet: Techniques in Differentiation – 2 | Quick Chain Rule | | Method: Finding Derivatives | | | | --- | --- | --- | --- | --- | | Logarithmic functions (base e only)
d af '( x)
(a log e f ( x) ) = , f ( x) > 0 dx f ( x)
dy Derivative of base numeral
= Coefficient of log × dx e Base numeral
Example: ddx~~ ~~ (−3loge (tan x) ) = − ~~ ~~3steacn2xx | | Step 1: Rewrite all terms as powers on x .
1
3 3
eg. x = x
Step 2: Bring terms involving x in the denominator to the top by changing the sign on the power.
eg.
32 x4
=
3x−4
2
Step 3: Simplify expressions.
Step 4: Differentiate.
Step 5: Re-write using positive powers.
| | | | | | | | | | Quick Chain Rule | | | | | | Trigonometric functions
ddx a sin ( f ( x) ) = af '( x) cos ( f ( x) ) ~~ ~~ddx~~ ~~ a cos ( f ( x) ) =−af '( x) sin ( f ( x) )
ddx~~ ~~ a tan ( f ( x) ) = af '( x) sec2 ( f ( x) )
i.e. dydx = Coefficient of trig × Derivative of the angle × Derived trigonometric function with the original angle
d 􀂧 1 3 􀂷 1 2 3 2 3
Example: dx 􀂩􀂨 3 cos( x + 2)􀂹􀂸 = 3 × 3x × − sin( x + 2) =−x sin( x + 2) | | | | | | | | | | | | Product Rule | | | | Specific Points | | The Product Rule states that given two functions (denoted as u and v ), then: ddx (u.v) = u. ~~ ~~dvdx + v. ~~ ~~dudx = ( First × Do Second ) + (Second × Do First)
d 2 2
Example: dx ( sin x. cos x ) = (sin x × − sin x) + (cos x × cos x) = sin x + cos x = 1 | | | | To find the

FAQs

What is the key idea behind the chain rule in this cheat sheet?

Identify the inside function u when y = f(u). Differentiate y with respect to u, then multiply by du/dx.

How does the quick chain rule handle powers on brackets?

It differentiates by multiplying the coefficient by the derivative of the bracket’s contents, then reduces the bracket power by one.

When should you use the product rule or quotient rule?

Use the product rule for products u·v and the quotient rule for division u/v (with v ≠ 0). Substitute into the appropriate formula to get the derivative.

Oct 13, 2026

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