Practise deriving dy/dx from implicit or parametric curves, simplifying it before substitution and using zero, undefined or reciprocal gradients for tangents and normals.
The parametric equations of a curve are
Obtain a simplified expression for dxdy in terms of t.
State d x/ d t=2 t-2/2 t+1
B1
Use correct quotient rule or correct product rule
M1
Might split first as 1/2-1/4 t+2.
Obtain d y/ d t=(2 t+1). 1-t(2)(2 t+1)^2 oe
A1
Simplifies to 1(2 t+1)^2
Use d y/ d x=d y/ d t / d x/ d t
M1
Marking guidance:
Allow for d x/ d y.
Obtain answer 1/2(2 t+1)(2 t-1)(t+1)
A1
OE. Must have 1 in numerator.
May have, e.g., 1(2 t+1)(4 t^2+2 t-2). ISW.
Obtain x=(y/1-2 y)^2- (2 y/1-2 y+1)
B1
OE, e.g. x=(y/1-2 y)^2- (1/1-2 y).
Use correct quotient rule to differentiate
M1
Obtain d x/ d y=(2 y/1-2 y)((1-2 y)+2 y(1-2 y)^2)-2/1-2 y
A1
OE, d y/ d x=(1-2 y)^32 y-2(1-2 y)^2=(1-2 y)^3-2(1-4 y)(1-y)d y/ d x=(1-2 t/2 t+1)^3-2(1-4 t/2 t+1)(1-t/2 t+1)=1/-2(2 t+1)(1-2 t)(t+1)
M1
Express in terms of t.
Obtain answer 1/2(2 t+1)(2 t-1)(t+1)
A1
OE