differentiation year 2

Question 1

shapes of functions

Answer 1

when dy/dx >0 function is increasing
when dy/dx<0 function is decreasing
when dy/dx =0 the function is stationary

Question 2

what happens dy/dx and d2y/dx2 when max

Answer 2

dy/dx=0
d2y/dx2 <0
point is max and the curve is concave

Question 3

what happens dy/dx and d2y/dx2 when min

Answer 3

dy/dx=0
d2y/dx2 >0
point is minimum and curve is convex

Question 4

points of inflection

Answer 4

at a point of inflection d2y/dx2 =0
however if dy/dx=0 further investigation is needed to determine the nature of the point
if this was the case you would then examine either side of the point, to determine if the second derivative is changing shape/ changing from neg to pos or pos to neg. If this is the case then the point is a point of inflection

Question 5

derivative of y=sinx

Answer 5

cosx

Question 6

derivative of y=cosx

Answer 6

-sinx

Question 7

derivative of e^x

Answer 7

e^x

Question 8

derivative of e^ax

Answer 8

ae^ax

Question 9

lim 0—–> sinx/x

Answer 9

1

Question 10

lim 0——-> 1-cosx/x

Answer 10

0

Question 11

derivative of y=Inx or y=Inax

Answer 11

1/x

Question 12

derivative if y=a^x

Answer 12

a^xIna

Question 13

product rule

Answer 13

dy/dx = uv’ + vu’
or = u dv/dx +v du/dx

can often expect to see chain rule needing to be used in product rule questions so need to look out for that

Question 14

quotient rule

Answer 14

given in formula booklet but may be easier to remember in this form
for y= u/v
y= v du/dx-u dv/dx
————————-
v^2

Question 15

parametric equations

Answer 15

an equation where the variables (usually x and y) are expressed in terms of a third parameter, usually expressed as t

Question 16

chain rule

Answer 16

for composite functions
dy/dx = dy/du x du/dx

• if you define a composite function as y= f(g(x)) then dy/dx= f’(g(x)) x g’(x)

Question 17

inverse functions

Answer 17

for x=f(y)
dy/dx= 1
——-
dx/dy

Question 18

differentiating parametric functions

Answer 18

can find dy/dx
x=f(t) and y=g(t) by using the chain rule
dy/dx= dy/dt x dt/dx

Question 19

cartesian equation

Answer 19

can be written in the form y=f(x)
It is an equation written only in terms of x and y