Y2, C9 - Differentiation

Question 1

Differential of cos(x)

Answer 1

-sin(x)

Question 2

Differential of sin(x)

Answer 2

cos(x)

Question 3

Derivative of sin(kx)

Answer 3

kcos(kx)

Question 4

Derivative of cos(kx)

Answer 4

-ksin(kx)

Question 5

Differential of a^x

Answer 5

lna * a^x

Question 6

Differential of a^kx

Answer 6

klna * a^kx

Question 7

Differential of lnx

Answer 7

1 / x

Question 8

Differential of ln(kx)

Answer 8

1 / x

Question 9

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

Answer 9

Chain rule

Question 10

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

Answer 10

Product rule

Question 11

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

Answer 11

The quotient rule

Question 12

What is the chain rule

Answer 12

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

Question 13

What is the shortcut of the chain rule

Answer 13

Differentiate as usual
Multiply by the differential of u

Question 14

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

Answer 14

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

Question 15

Mentally differentiate 3(8 - x)^-6

Answer 15

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

Question 16

Mentally differentiate e^(x^2 + x)

Answer 16

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

Question 17

Mentally differentiate (2^x + 1)^2

Answer 17

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

Question 18

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

Answer 18

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

Question 19

Differentiate ln(x^3)

Answer 19

3x^2 / x^3
3 / x

Question 20

What is the reciprocal of dy / dx

Answer 20

1 / (dx / dy)

Question 21

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

Answer 21

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

Question 22

What is the product rule

Answer 22

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

Question 23

Differentiate ln(3x)

Answer 23

1 / x

Question 24

Differential of ae^bx

Answer 24

abe^bx

Question 25

What is the quotient rule

Answer 25

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

Question 26

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

Answer 26

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

Question 27

What is the differential of tan(kx)

Answer 27

k sec^2(kx)

Question 28

What is the differential of sec(kx)

Answer 28

k sec(kx) * tan(kx)

Question 29

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

Answer 29

Quotient rule

Question 30

What is the differential of cot(x)

Answer 30

-cosec^2(x)

Question 31

What is the differential of cosec(x)

Answer 31

-cosec(x) * cot(x)

Question 32

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

Answer 32

sec^2(x)

Question 33

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

Answer 33

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))

Question 34

What is the equation for parametric differentiation

Answer 34

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

Question 35

What is the difference between explicit and implicit functions

Answer 35

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

Question 36

Differentiate y^2 with respect to x

Answer 36

= 2y * dy/dx

Question 37

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

Answer 37

dy / dx

Question 38

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

Answer 38

2x - siny * (dy/dx)

Question 39

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

Answer 39

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

Question 40

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

Answer 40

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

Question 41

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

Answer 41

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

Question 42

What way is a curve swerving if it is concave

Answer 42

Right

Question 43

What way is a curve swerving if it is convex

Answer 43

Left

Question 44

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

Answer 44

0

Question 45

When is f(x) convex

Answer 45

f''(x) > 0

Question 46

When is f(x) concave

Answer 46

f''(x) < 0

Question 47

What does the rate of something mean

Answer 47

How it changes per time (seconds)

Question 48

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

Answer 48

cm^2 * s^-1

Question 49

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

Answer 49

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

Question 50

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

Answer 50

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

Question 51

How do you solve differential equations

Answer 51

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

Question 52

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

Answer 52

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

Question 53

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

Answer 53

ln k

Question 54

How do you solve differential equations with boundary conditions

Answer 54

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

Question 55

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

Answer 55

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