Challenging +1.2 This question requires applying the first principles definition of differentiation to find f''(x) from f'(x), which is more demanding than routine differentiation. However, it's a straightforward application of a standard technique with a simple polynomial, requiring careful algebraic manipulation but no novel insight. The 5 marks reflect moderate length rather than exceptional difficulty.
Question 10:
10 | States
5 ( x + h ) 3 + ( x + h ) − ( 5 x 3 + x )
Condone missing brackets | 1.1a | M1 | f ( x + h ) − f ( x ) 5 ( x ( 3) ( ) + + + − h x h 5 x 3 + x )
=
h h
5 ( x 3 2 2 + + + x h x h h 3 3 3 ) + h − 5 x 3
=
h
1 x 5 2 2 3 + + + h x h h 1 5 5 h
=
h
= 1 x 5 2 2 + + + x h h 1 5 5 1
f ( x ) = lh i m→ 1 5 x 2 + 1 5 x h + 5 h 2 + 1
0
f ( x ) = 1 5 x 2 + 1
Expands ( x+h )3 correctly
Like terms need not be collected
Accept all terms multiplied by 5
May be embedded | 1.1a | M1
1 5 x 2 h + 1 5 x h 2 + 5 h 3 + h
Obtains
h
correctly eliminating 5x3 and x
PI by 1 5 x 2 + 1 5 x h + 5 h 2 + 1
having seen a division by h
Like terms need not be collected | 1.1b | A1
Obtains 1 5 x 2 + 1 5 x h + 5 h 2 + 1
by correctly dividing by h
Like terms need not be collected | 2.1 | A1
Completes a reasoned
lim
argument using the to
h→0
prove that f ( x ) = 1 5 x 2 + 1
f( )
x may be seen on the final
line or before
dy
f( )
Do not allow for x
dx | 2.5 | R1
Question 10 Total | 5
Q | Marking instructions | AO | Marks | Typical solution
It is given that
$$f'(x) = 5x^3 + x$$
Use differentiation from first principles to prove that
$$f''(x) = 15x^2 + 1$$
[5 marks]
\hfill \mbox{\textit{AQA Paper 3 2024 Q10 [5]}}