Differentiation

Question 1

dy/dx is th same as

Answer 1

1/dx/dy

Question 2

the derivative of e^x is

Answer 2

e^x

Question 3

derivative of lnx

Answer 3

1/x

Question 4

line point (x1,yy1) with gradietn m, equation

Answer 4

y-y2=m(x-x1)

Question 5

y=uv derivaive=

Answer 5

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

Question 6

y= u/v derivative

Answer 6

dy/dx = ((v)du/dx -(u)dv/dx)/v²

Question 7

sinx DIFFERENTIATION order

Answer 7

sinx —> cosx —> -sinx —> -cosx (—> sinx)

Question 8

derivative of tanx

Answer 8

sec²x

Question 9

chain rule

Answer 9

dy/dx=dy/dx x du/dx

Question 10

derivative of f(ax)

Answer 10

af’(ax)

Question 11

derivative of f(ax+b) is

Answer 11

af’(ax+b)

Question 13

a curve with parametric equation x=t + 1/t, y= t- 1/t, t>0

a) find the tangent where t=2, write in form ax+by=c
b) find the equation of the normal where t=2

Answer 13

dx/dt = 1- 1/t², dy/dt =1+ 1/t²

dy/dx=dy/dt/dx/dt =1+(1/t²)/1-(1-t²)

when t=2, dy/dx =5/3

when t=2 the coordinates are (2+(1/2),2-(1/2)) or (5/2,3/2)

gradient of tangent is 5/3 so the equation

y- (3/2) =(5/3)(x-(5/2))

5x-3y=8

b)gradient of normal = -3/5

y-(3/2)=-(3/5)(x-(5/2))

3x+5y=15

Question 14

differentiate xy+y=9 with respect to x

Answer 14

(x)dy/dx+y+dy/dx=0

Question 15

differentiate 2x²+xy+y²=28

find equations of tangent and normal at (3,2)

Answer 15

dy/dx=4x +((x)dy/dx +y) +(2y)dy/dx =0

x=3 and y=2

4(3) +(3dy/dx +2) +4dy/dx =0

dy/dx(3+4)=-14

dy/dx=-2

tangent gradient at (3,2) =-2

2x+y-8=0

gradient of normal at (3,2) is 1/2

x-2y+1=0

Question 16

dy/dx of y²

Answer 16

2y(dy/dx)

Question 17

find the tangent and normal of x²y +xy²=12 at (3,1)

Answer 17

[x²(dy/dx) + 2yx] + [x2y(dy/dx) + y²] = 0

[3²(dy/dx) + 2(3)(1)] + [(3)(2)(1)(dy/dx) + 1²] = 0

9(dy/dx) + 6(dy/dx) =-7

dy/dx= -7/15

tangent: (y-1=-7/5(x-3))—> 15y+ 7x-36=0
normal: (y-1= 15/7(x-3)) —> 7y+15x+38=0