Moderate -0.5 This is a straightforward application of the chain rule with the standard derivative of arcsin. While it's a Further Maths topic (making it slightly harder than typical A-level), it requires only direct recall of the formula d/dx[sin⁻¹(u)] = 1/√(1-u²) and basic chain rule application, making it easier than average overall.
Allow any equivalent correct form e.g. 2 x (1 − x 4 ) − 0 .5 .
Must be in terms of x.
Condone ( x 2 ) 2 for x4.
ISW once correct answer seen.
Alternative method
dy d x
cosy =2x or c o s y = 2 x
Answer
Marks
Guidance
dx d y
B1
B1
d y
1 − ( x 2 ) 2 = 2 x
Answer
Marks
Guidance
d x
M1
Replacing c o s y with 1(x2)2 in their derivative of the form
d y
c o s y = 2 x (or equivalent if differentiating with respect to y).
d x
d y 2 x
=
Answer
Marks
Guidance
d x 1 − x 4
A1
Allow any equivalent correct form e.g. 2 x (1 − x 4 ) − 0 .5 .
Must be in terms of x.
Condone ( x 2 ) 2 for x 4 .
ISW once correct answer seen.
[3]
Allow any equivalent correct form e.g. 2 x (1 − x 4 ) − 0 .5 .
Must be in terms of x.
Condone ( x 2 ) 2 for x 4 .
ISW once correct answer seen.
Question 1:
1 | 1
(1−(x2)2
2 x
d y 2 x
=
d x 1 − x 4 | B1
M1
A1
[3] | 1.1
1.1
1.1 | 1
For seen.
(1−(x2)2
1 1
For 2 x f ( x ) where f ( x ) = or ONLY.
1 − ( x 2 ) 2 1−x2
Allow any equivalent correct form e.g. 2 x (1 − x 4 ) − 0 .5 .
Must be in terms of x.
Condone ( x 2 ) 2 for x4.
ISW once correct answer seen.
Alternative method
dy d x
cosy =2x or c o s y = 2 x
dx d y | B1 | B1 | For correctly differentiating implicitly with respect to either x or y. | For correctly differentiating implicitly with respect to either x or y.
d y
1 − ( x 2 ) 2 = 2 x
d x | M1 | Replacing c o s y with 1(x2)2 in their derivative of the form
d y
c o s y = 2 x (or equivalent if differentiating with respect to y).
d x
d y 2 x
=
d x 1 − x 4 | A1 | Allow any equivalent correct form e.g. 2 x (1 − x 4 ) − 0 .5 .
Must be in terms of x.
Condone ( x 2 ) 2 for x 4 .
ISW once correct answer seen.
[3]
Allow any equivalent correct form e.g. 2 x (1 − x 4 ) − 0 .5 .
Must be in terms of x.
Condone ( x 2 ) 2 for x 4 .
ISW once correct answer seen.