9. Differentiation

Question 1

Concave functions

Answer 1

The second derivative is <= 0 for all x, any two points on the curve joined by a line are below the curve

Rise to a maximum and then fall again

Question 2

Convex functions

Answer 2

The second derivative is >= 0 for all x, any two points on the curve joined by a line are above the curve

Fall to a minimum and then rise again

Question 3

Chain rule process

Answer 3

1. State the original function
2. State what your temporary variable is (u = inside of brackets)
3. Set y to u^power and do dy/du
4. Find du/dx
5. dy/dx = dy/du x du/dx and replace u with the inside of the brackets

Question 4

Chain rule f(x)

Answer 4

Write y = f(x) and then put f’(x) at the end

Question 5

sin(x) differentiated

Answer 5

cos(x)

Question 6

cos(x) differentiated

Answer 6

-sin(x)

Question 7

Differentiating sin(x) and cos(x) by first principles

Answer 7

Use the angle addition formulae and use sin(h)/h and 1-cos(x)/h = 0 by finding them with small angles

Question 8

e^x differentiated

Answer 8

e^x

Question 9

ln(x) differentiated

Answer 9

1/x

Question 10

sin(h)/h

Answer 10

1

Question 11

1 - cos(h)/h

Answer 11

0

Question 12

Product rule

Answer 12

Used where y = uv where u and v are two functions

1. Differentiate each with respect to x
2. Multiply by the other function
3. Sum

Question 13

e^(ax + b) differentiated

Answer 13

a x e^(ax+b)

Question 14

ln(x^2) differentiated

Answer 14

2x
—
x^2

Question 15

ln(ax+b) differentiated

Answer 15

a
———
ax + b

Question 16

cos(ax) differentiated

Answer 16

-a sin (ax)

Question 17

sin(ax) differentiated

Answer 17

a cos(ax)

Question 18

Quotient Rule uses

Answer 18

where y can be written as u/v where u,v are functions

Question 19

Quotient rule

Answer 19

.      du         dv
    v ----  -  u -----
dy   dx         dx
---- = ----------------
dx            v^2

Question 20

Tan x differentiated

Answer 20

Sec^2x

Question 21

Explicit Equations

Answer 21

Where y can be written as f(x)

Question 22

Implicit equations

Answer 22

Not written in the form y = f(x)

E.g. x^2 + y^2 = 5, y + 7 = 3x

Question 23

d/dx (f(y))

Answer 23

f’(y) x dy/dx

dy/dx won’t be in terms of numbers

Question 24

Implicit differentiation steps

Answer 24

1. Write d/dx(LHS) = d/dx(RHS)
2. Differentiate each side leaving the derivative of y as dy/dx
3. Rearrange and factor out dy/dx
4. Divide for dy/dx=

Question 25

d/dx(xy)

Answer 25

y + x(dy/dx)

Question 26

d/dx a^(bx+d)

Answer 26

b ln(a) a^bx+d

Question 27

sec(x) differentiated

Answer 27

sec(x)tan(x)

Question 28

cosec(x) differentiated

Answer 28

-cosec(x) cot(x)

Question 29

y = f(g(x)) differentiated (chain rule)

Answer 29

g’(x) f’(g(x))

Question 30

Differentiating inverse trig

Answer 30

Rearrange for x = a function of y
Find dx/dy
Find the reciprocal for dy/dx
Substitute to be in terms of x

Question 31

Connected rates of change

Answer 31

Use the chain rule to write the derivative you need as a product of two derivatives
You may need to differentiate an equation for area or volume and do 1/ a derivative

Question 32

a^x differentiated

Answer 32

ln(a) a^x