Differentiation

What can differentiation also be called

• rate of change
• speed or velocity of an object
• m or slope of the tangent

How is the derivative of y expressed

dy
dx

How is the derivative of f(x) expressed

f¹(x)

What are the 5 mathematical families?

Logarithms
Inverses
Algebraic
Trigonometric
Exponential

How do you differentiate
y = 16x⁵

multiply the number by the power
reduce the power by one
apply the power to x
80x⁴

How do you differentiate
5
x³

bring x on top and make its power negative
multiply the number by the power
reduce the power by one
apply the power to x
make x the denominator again
-15
x⁴

How do you differentiate
Sin4x

keep the angle (number directly after Sin)
get the derivative of Sin (log tables)
multiply by the derivative of the angle
(steps also apply for Cos and Tan)
Cos4x × 4

How do you differentiate
12Cos3x

keep the angle (number directly after Cos)
keep the coefficient
get the derivative of Cos (log tables)
multiply by the derivative of the angle
(steps also apply for Sin and Tan)
12 × -Sin3x × 3
-36Sin3x

How do you differentiate
Sin9x
16

keep the angle (number directly after Sin)
keep the denominator
get the derivative of Sin (log tables)
multiply the top by the derivative of the angle
(steps also apply for Sin and Cos)
Cos9x × 9
16

How do you differentiate
eˣ

keep everything
multiply term by the derivative of the power
eˣ is unusual as it is the only function that stays the same when differentiated

How do you differentiate
100e⁴ˣ

keep everything
multiply term by the derivative of the power
100e⁴ˣ × 4

How do you differentiate
6e²⁰ˣ
19

keep everything
multiply the top by the derivative of the power
6e²⁰ˣ × 20
19

How do you differentiate
lnx³

put 1 over the angle
multiply by the derivative of the angle

1 × 3x² = 3
x³ x

How do you differentiate
logₑ√3x-1

express logₑ without the base (ln)
turn the root into a power
using log law, move the power to the front of the term
1/2ln (3x-1)
keep the front term
put 1 over the angle
multiply by the derivative of the angle
3 × 1
2 3x -1

How do you differentiate
Sin⁻¹6x

use formula in log tables
if no denominator, let a = 1
let x = 6x
1
√1² - 36x²

How do you differentiate
Tan⁻¹10x

use formula in log tables
if no denominator, let a = 1
let x = 10x
1
1² + 100x²

When is the product rule used

when differentiating two x terms that are being multiplied together

Explain the product rule in words

T1 × derivative of T2 + T2 × derivative of T1

Find the derivative of
(3x + 6)(5x - 9)

(3x + 6)(5) + (5x - 9)(3)

Find the derivative of
Sin4x × e³ˣ

Sin4x × e³ˣ × 3 + e³ˣ × Cos4x × 4

Find the derivative of
ln2x × 1
√x

ln2x × x⁻¹/²
ln2x × ½x⁻³/² + x⁻¹/² × 1/2x × 2

When is the quotient rule used

when differentiating a fraction with x terms on the top and bottom

How is the quotient rule used

Find the derivative of
11x - 4
9x + 3

(9x + 3)(11) - (11x - 4)(9)
(9x + 3)²
69
(9x + 3)²

When is the chain rule used

when differentiating a number being multiplied by a bracket in which there is an x term

How do you differentiate
4(5x - 1)⁸
using the chain rule

multiply the number by the power
keep brackets
lower the power by 1
multiply term by the derivative of the brackets

How do you differentiate
Cos³x

(Cosx)³
3(Cosx)² × -Sinx × 1

How do you differentiate
√x - 4

(x - 4)¹/²
½(x - 4)⁻¹/² × 1
1 × 1
2 x - 4

How do you differentiate
1
(x + 3)⁻⁴

rewrite equation not as a fraction