6 Differentiation

Question 1

how to differentiate y=x^n

Answer 1

dy/dx = nx^(n-1)

Question 2

if f’(a) >0

Answer 2

when x=a, f is increasing

Question 3

if f’(a) >0

Answer 3

when x=a, f is decreasing

Question 4

if f(x)=3e^x -1/x show f(x) in increasing for x>0

Answer 4

f’(x) = 3e^x + 1/x^2

x>0 so e^x >0 and 1/x^2 >0
so 3e^x +1/x^2 >0

Question 5

if f(x) =lnx what is f'(x)=
and why

Answer 5

1/x

beacause its gradient is the reciprocal of y=e^x

Question 6

dx/dy=

Answer 6

1/dy/dx

Question 7

Ln3 what is dy/dx

Answer 7

1/0

Question 8

Inverse of f(x) = e^x + 1

Answer 8

X=e^y + 1

Ln(x-1) = f^-1(x)

Question 9

F(x) = ln3x -6

F^-1(x)=

Answer 9

Y+6 =ln(3x)

=1/3(e^x+6)

Question 10

G(x) = 2e^(x-5) +3

G^-1(x) =

Range And domain

Answer 10

G^-1(x) = ln(1/2(x-3)) +5

Gs range= g(x) >3
G-1s domain = g-1 is x>3
Domain oF g(x) = xER
Range of g-1 — g-1 ER

Question 11

F(x)= 5x^4 -ln(5x^4)

F’(x)=

Answer 11

F’(x)= 20x^3 - 4/x

Question 12

Find stationary of y= 8x^2 -ln(x)

Answer 12

Dy/dx =0

X=+- 1/4

y= 8/16 -ln1/4 =1.886
-1/4 doses not exist because ln(-) is not possible

S.p (0.25, 1.886)

D2y/dx2 = 16 + 1/x^2
x=0.25
16 + 1/0.25^2 =32

32> 0 so minimum point

Question 13

Product rule

Answer 13

Y can be split into u and v
U Du/dx
V. Dv/dx
Cross multiply

V du/dx + u dv/dx

Question 14

differentiate y= x^1/2 lnx

Answer 14

u=x^1/2, du/dx = 1/2x^-1/2
v=lnx, du/dx=1/x

rearranged to x^-1/2(1/2lnx+1)

Question 15

stationary points of x^2.e^x

find its type

Answer 15

dy/dx= e^x(2+x)x

dy/dx=0
e^x never equals 0
x=0
x=-2

x=0, y=0
x=-2, y=4e^-2

d2y/dx2 = e^x(4x+x^2+2)
x=0, d2y/dx2=2, 2>0 minimum
x=-2, d2y/dx2 =4e^-2, 4w^-2<0 maximum

Question 16

quotient rule, how do you get there

Answer 16

product rule

du/dx =ydv/dx +vdy/dx

if y=u/v then you can rearange

dy/dx = (du/dx -(u/v)dv/dx)v

=(vdu/dx -udv/dx)v^2

Question 17

when do you use the quotient rule

Answer 17

for fractions

Question 18

y= (1+x^2) / (1+e^x) differentiate

Answer 18

y=u/v

2x+e^x(2x-1+x^2) /(1+e^x)^2 /(1+e^x)^2

Question 19

how to find the gradient of y=sinx

Answer 19

x point you take is h
height of graph is thus sinh
and the x value is just h

gradient is sinh/h
make sure you’re in radians

Question 20

f(x)= e^x(sinx) differentiate

Answer 20

e^x(sinx +cosx)

Question 21

f(x)= tanx/(x^2 +1) =u/v

Answer 21

sec^2(x^2+1) -2xtanx /(x^2+1)^2

Question 22

differentiate tanx

Answer 22

tanx =sinx/cosx
use quotint rule

cos^2x +sin^2x /cos^2x

1+tan^2x =sec2x

Question 23

y=sin(x^2 -3) differentiat

Answer 23

is a function of a function
so you use chain rule du/dx x dv/dx

y=sinu, u= x^2 -3
dy/du= cosu, du/dx= 2x

dy/dx = 2xcos(x^2 -3)

Question 24

differentiating sinx (all the others)

Answer 24

sinx > cosx > -sinx > -cosx > sinx

Question 25

y= sqqrt(x^2 +1) differentiate

Answer 25

chain rule

dy/dx = x/sqqrt(x^2 +1)

Question 26

chain rule

Answer 26

dydu x du/dx

Question 27

y= ln(1+x^2) differentiate

Answer 27

chain rule

2x/(1+x^2)

Question 28

extended chain rule

Answer 28

dy/dx = dy/du x du/dv x dv/dx

Question 29

differentiate sqqrt(sinx^2)

Answer 29

ends up 1/2(sinx^2)^1/2 x cosx^2 x 2x

xcosx^2/sqqrt(sinx^2)

Question 30

chain used it

Answer 30

somthing inside something

Question 31

quotient used if

Answer 31

equation

Question 32

product used if

Answer 32

used when its something x something

Question 33

differentiate x^2e^x + 3lnx

Answer 33

e^x(2x+x^2) +3/x

Question 34

y= x^2.e^-3x where x=1

find point

Answer 34

dy/dx= e^-3x(2x-3x^2)

x=1 e^-3(-1) =-e^-3

Question 35

derivative of f(ax) is

Answer 35

af’(ax)

Question 36

derivative of f(ax+b) is

Answer 36

af’(ax+b)