Differentiation

Question 1

Differentiate xⁿ

Answer 1

nx^n-1

Question 2

What is the derivative of e^x?

Answer 2

e^x

Question 3

What is the derivative of e^kx?

Answer 3

ke^kx

Question 4

What is the derivative of e^f(x)?

Answer 4

f’(x)e^f(x)

Question 5

What is the derivative of lnx?

Answer 5

1 / x

Question 6

What is the derivative of ln(ax)?

Answer 6

1 / x

Question 7

What is the derivative of ln f(x) ?

Answer 7

f’(x) / f(x)

Question 8

What is the derivative of sinx?

Answer 8

cos x

Question 9

What is the derivative of sin kx?

Answer 9

kcos kx

Question 10

What is the derivative of cos x?

Answer 10

-sin x

Question 11

What is the derivative of cos kx?

Answer 11

-ksin kx

Question 12

What is the derivative of tanx?

Answer 12

sec²x

Question 13

What is the derivative of tan kx?

Answer 13

ksec² kx

Question 14

What is the derivative of cosec x?

Answer 14

-cosecx cotx

Question 15

What is the derivative of cosec kx?

Answer 15

-k coseckx cot kx

Question 16

What is the derivative of sec x?

Answer 16

secx tanx

Question 17

What is the derivative of sec kx?

Answer 17

kseckx tankx

Question 18

What is the derivative of cotx?

Answer 18

-cosec²x

Question 19

What is the derivative of cot kx?

Answer 19

-k cosec²kx

Question 20

What is the derivative of a^x?

Answer 20

a^x lna

Question 21

What is the derivative of a^kx?

Answer 21

a^kxk lna

Question 22

What is the derivative of arcsin x?

Answer 22

1 / √(1-x²)

Question 23

What is the derivative of arccos x?

Answer 23

-1 / √(1-x²)

Question 24

What is the derivative of arctan x?

Answer 24

1 / (1+x²)

Question 25

Find the derivative of sinx from first principles

Answer 25

ONLY in radians

f(x) = sin x

f’(x) = h→0 (f(x+h) -f(x)) / h

f’(x) = h→0 (sin(x+h) - sin(x)) / h

f’(x) = lim h→0 (sinxcosh + cosxsinh - sinx) / h

f’(x) = lim h→0 (sinxcosh - sin x) / h + cosxsinh / h

f’(x) = sinx(cosh - 1) / h + cosx (sinh) / h

(cosh -1) / h → 0 and sinh / h → 1

f’(x) = lim h→0 (sin(x+h)-sinx) / h = cos x

Question 26

Find the derivative of cosx from first principles?

Answer 26

f(x) = cosx

f’(x) = (cos(x+h) - cosx) / h

f’(x) = h → 0 (cosxccosh - sinxsinh - cosx) / h

f’(x) = h → 0 (cosxcosh - cosx / h) - (sinxsinh / h)

f’(x) = lim h → 0 cosx (cosh-1 / h) - sinx (sinh/h)

cosh -1 /h → 0 and sinh/h → 1

f’(x) = 0-sinx = -sinx

Question 27

What is the derivative of lnxⁿ?

Answer 27

n lnx

Question 28

What is the chain rule?

Answer 28

dy/dx = du/dx * dy/du

Do this when there is a function of a function

Question 29

Use the chain rule to differentiate (x+3)⁷

Answer 29

u = x + 3 and du/dx = 1

y = u⁷ and dy/du = 7u⁶

dy/dx = 7u⁶ = 7(x+3)⁶

Question 30

What is the product rule?

Answer 30

y = uv

dy/dx = (u*dv/dx) + (v*du/dx)

Question 31

What is the quotient rule?

Answer 31

y = u / v

dy/dx =(v*du/dx - u*dv/dx ) / v²

Question 32

What is implicit differentiation?

Answer 32

Equations which emphasise x and y as equal partners

d/dx f(y) = f’(y) * dy/dx

e.g. d/dx (y²) = 2y * dy/dx

Question 33

When is a function concave at a given interval?

Answer 33

When f’‘(x) <= 0 for every value of x in the interval

This is when the gradient is decreasing (maximum)

Question 34

When is a function convex at an interval?

Answer 34

Only when f’‘(x) >= 0 for every value of x

This is when the gradient is increasing (minimum)

Question 35

What is a point of inflection in reference to convex and concave?

Answer 35

Point of inflection is where f’‘(x) changes sign
You need to show f’‘(x) = 0 at the point and that there are different signs on either side

Concave on one side and convex on the other

Question 36

What does the interval [a,b] mean?

Answer 36

a <= x <= b

Question 37

How do you differentiate parametric equations?

e.g. x = 2at² and y = 4at

Answer 37

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

Question 38

What does the expression “increasing at the rate of” mean?

Answer 38

Implies differentiation with respect to time

Question 39

What is a differential equation?

Answer 39

An equation which involves a rate of change

Question 40

When proportional is mentioned in the question, what should the equations show?

e.g. x is proportional to y

and when decreasing the rate of change (decay)

Answer 40

x = ky

if decay/decrease then x = -ky