differentiation

Question 1

find dy/dx

y = ln6x⁹

Answer 1

simplify to get an expression in terms of lnx

ln6x⁹ = 9ln6x= 9(lnx +ln6) = 9lnx + 9ln6

dy/dx = 9/x

Question 2

find dy/dx

y= sin²x

Answer 2

this is a composite function (a function of a function) so we apply the chain rule

let u = sinx so y = u²

du/dx = cosx

dy/du = 2u

dy/dx = dy/du x du/dx = 2u cos x

substitute u = sinx back in:

dy/dx = 2sinxcosx

Question 3

differentiate y=tanx

Answer 3

Question 4

prove that if y=secx dy/dx = secxtanx

Answer 4

Question 5

prove that if y = cosecx, dy/dx= - cosecx cotx

Answer 5

Question 6

prove that if y=cotx, dy/dx=-cosec²x

Answer 6

Question 7

if y=arcsinx find dy/dx

Answer 7

Question 8

if y=arcosx find dy/dx

Answer 8

Question 9

if y = arctanx find dy/dx

Answer 9

Question 10

what does it mean if a tangent is parallel to the y-axis

Answer 10

has infinite gradient so if dy/dx is in the form u/v, the denominator = 0

dx/dy=0