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A Level Maths Pure > Y2, C9 - Differentiation > Flashcards

Y2, C9 - Differentiation Flashcards

1
Q

Differential of cos(x)

A

-sin(x)

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2
Q

Differential of sin(x)

A

cos(x)

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3
Q

Derivative of sin(kx)

A

kcos(kx)

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4
Q

Derivative of cos(kx)

A

-ksin(kx)

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5
Q

Differential of a^x

A

lna * a^x

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6
Q

Differential of a^kx

A

klna * a^kx

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7
Q

Differential of lnx

A

1 / x

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8
Q

Differential of ln(kx)

A

1 / x

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9
Q

How do you differentiate a composite function (f(g(x))

A

Chain rule

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10
Q

How do you differentiate the product of two functions (y = x * sin2x)

A

Product rule

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11
Q

How do you differentiate the division (or quotient) of two functions

A

The quotient rule

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12
Q

What is the chain rule

A

dy / dx = (dy / du) * (du / dx)

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13
Q

What is the shortcut of the chain rule

A

Differentiate as usual
Multiply by the differential of u

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14
Q

Use the shortcut of the chain rule to differentiate (3x^4 + x)^5

A

5(3x^4 + x)^4 * (12x^3 +1)

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15
Q

Mentally differentiate 3(8 - x)^-6

A

-18(8 - x)^-7 * (-1) =
18(8 - x)^-7

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16
Q

Mentally differentiate e^(x^2 + x)

A

e^(x^2 + x) * (2x + 1)

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17
Q

Mentally differentiate (2^x + 1)^2

A

2(2^x + 1) * (ln2 * 2^x)

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18
Q

Mentally differentiate y = -sin^-2(x)

A

y = - (sinx)^-2
–> 2sin^-3(x) * cosx
–> 2cos(x)sin^-3(x)
–> 2cos(x)cosec^3(x)

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19
Q

Differentiate ln(x^3)

A

3x^2 / x^3
3 / x

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20
Q

What is the reciprocal of dy / dx

A

1 / (dx / dy)

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21
Q

What is dy / dx when x = 2y^2 + y

A

dx / dy = 1 / (dy / dx)
dx / dy = 4y + 1
dx / dy = 1 / (4y + 1)

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22
Q

What is the product rule

A

If y = uv then dy/dx = u(dv/dx) + v(du/dx)
OR
y’ = uv’ + vu’

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23
Q

Differentiate ln(3x)

A

1 / x

24
Q

Differential of ae^bx

A

abe^bx

25
Q

What is the quotient rule

A

If y = u / v, then dy / dx =
(vu’ - uv’) / v^2

26
Q

Find the stationary point of y = sinx / e^2x

A

Quotient rule: dy / dx = (cosx - 2sinx) / e^2x
Multiply out by e^2x (e^2x = 0 has no solutions)
Divide remaining terms by tanx
1 - 2tanx = 0
1/2 = tanx
x = 0.464, y = 0.177

27
Q

What is the differential of tan(kx)

A

k sec^2(kx)

28
Q

What is the differential of sec(kx)

A

k sec(kx) * tan(kx)

29
Q

What rule would you use to differentiate tan, sec, cot, cosec

A

Quotient rule

30
Q

What is the differential of cot(x)

A

-cosec^2(x)

31
Q

What is the differential of cosec(x)

A

-cosec(x) * cot(x)

32
Q

What is 1 + tan^2(x) equal to

A

sec^2(x)

33
Q

Given that x = 2 sin(y), express dy/dx in terms of x

A

dx / dy = 2cos(y)
dy / dx = 1 / 2cos(y)
cos(y) = (root(4-x^2) / 2)
Sub in for cos(y)
dy / dx = 1 / (root(4-x^2))

34
Q

What is the equation for parametric differentiation

A

dy /dx = (dy / dt) / (dx / dt)

35
Q

What is the difference between explicit and implicit functions

A

Explicit functions have a variable as the subject
Implicit functions do NOT have a variable as the subject

36
Q

Differentiate y^2 with respect to x

A

= 2y * dy/dx

37
Q

In general when differentiating a function of y, but with respect to x, what should you multiply by

A

dy / dx

38
Q

Differentiate x^2 + cos(y) with respect to x

A

2x - siny * (dy/dx)

39
Q

Find dy/dx in terms of x and y where x^3 + x + y^3 + 3y = 6

A

3x^2 + 1 + (3y^2 + 3)dy/dx = 0
dy/dx = (-3x^2 - 1) / (3y^2 + 3)

40
Q

What does it mean if a tangent is parallel to the x axis

A

The gradient is 0
dy/dx = num / denom
num = 0

41
Q

What does it mean if the tangent is parallel to the y axis

A

The gradient is undefined
dy/dx = num / denom
denom = 0

42
Q

What way is a curve swerving if it is concave

A

Right

43
Q

What way is a curve swerving if it is convex

A

Left

44
Q

What is f’‘(x) equal to at a point of inflection

A

0

45
Q

When is f(x) convex

A

f’‘(x) > 0

46
Q

When is f(x) concave

A

f’‘(x) < 0

47
Q

What does the rate of something mean

A

How it changes per time (seconds)

48
Q

What would the unit of measurement be for dA / dt where t is time and A is area

A

cm^2 * s^-1

49
Q

How would you find the equation for the rate of change of volume of a sphere if you know the rate of change of radius

A

dV / dt = (dV/dr) * (dr/dt)
V = 4/3 * pi * r^3
dV/dr = 4 * pi * r^2
Therefore dV/dt = 4pi*r^2 * dr/dt

50
Q

If you know V in terms of r, S in terms of r and dV/dt. How would you calculate dS/dt.
V = volume
S = surface area
r = volume

A

Find dV/dr, dS/dr, dV/dt
dS/dt = (dS/dr) * (dr/dV) * (dV/dt)
Sub in known values and find dS/dt in terms of r

51
Q

How do you solve differential equations

A

Getting y in terms of x with NO dy/dx
Get y on LHS (possibly factorising first)
Then integrate both sides

52
Q

Find the general solution to dy/dx = xy + y

A

dy/dx = y(x+1)
(1/y) * (dy/dx) = x + 1
(1/y) dy = (x+1) dx
lny = 0.5x^2 + x + c
y = Ae^0.5x^2 + x
where A = e^c

53
Q

When solving differential equations, if your ‘+c’ becomes an ‘ln’, what should you call it

A

ln k

54
Q

How do you solve differential equations with boundary conditions

A

Sub in the given values to find the value of ‘c’ and then rewrite in general solution

55
Q

The rate of increase of a population is proportional to current population, form a differential equation and find its general solution

A

dP/dt = kP
(1/P) * dP/dt = k
ln(P) = kt + c
P = Ae^kt

A Level Maths Pure (18 decks)

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