Reciprocal of dy/dx & derivative of trig functions

Question 1

What does dy/dx mean?

Answer 1

the derivative of y with respect to x

Question 2

What does dx/dy mean?

Answer 2

the derivative of x with respect to y

Question 3

How can you work out dy/dx ?

Answer 3

dy/dx = 1/(dx/dy)

Question 4

x=2y^2 +y,
a) find dy/dx

b)find the eqn of tangent to the curve at the point where y=1

Answer 4

x=2y^2 +y
dx/dy = 4y + 1
dy/dx = 1/(dx/dy) = 1/4y+1

b) when y=1, x=2(1)^2+1 = 3 –> at point(3,1)
sub y=1 into dy/dx = 1/4(1) +1 = 1/5
therefore m of tangent=1/5
y-y1=m(x-x2)
y-1=1/5(x-3)

Question 5

How can you rewrite arcsin x ?

Answer 5

arcsinx = Sin-1 x

Question 6

How can you rewrite arcCos x?

Answer 6

arcCosx =Cos-1 x

Question 7

How do you differentiate y=arcsin x?

Answer 7

y=arcsinx –> y=Sin-1 x
Siny = Sin(Sin-1 x)
Siny=x
dx/dy=cosy
dy/dx=1/Cos y
BUT sin^2y + Cos^2y = 1
Cos^2y=1 - Sin^2y
Cos y = √1- Sin^2y
x=sin y so √1- Sin^2y becomes √1- x^2
therefore dy/dx = 1/Cosy =√1- x^2

Question 8

When differentiating arcSinx what is it important to know?

Answer 8

y=Sin-1x
Sin y = x
x=Sin y

Question 9

When differentiating arcCos x, what is it important to know?

Answer 9

y=Cos-1x
Cos y = x
x= Cos y

Question 10

When differentiating arctan x, what is important to know?

Answer 10

y=tan-1x
–> tany=x
x=tan y

Question 11

What is the derivative of y=Sin-1 x ?
(arcsin x)

Answer 11

dy/dx = 1/√1-x^2

Question 12

What is the derivative of y=Cos-1x ?
(arcCos x)

Answer 12

dy/dx = -1/√1-x^2

Question 13

What is the derivative of y=tan-1 x?
(arctan x) ?

Answer 13

dy/dx = 1/1+x^2