Challenging +1.2 This is a structured induction proof requiring differentiation of arctan and the product/quotient rule. While it involves Further Maths content and requires careful algebraic manipulation across multiple steps, the framework is standard: verify base case, assume for n=k, prove for n=k+1 using the product rule. The polynomial degree claim adds mild complexity but follows naturally from the differentiation. More routine than typical Further Maths proof questions requiring novel insight.
Prove by mathematical induction that, for all positive integers \(n\),
$$\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(\tan^{-1}x\right) = P_n(x)\left(1+x^2\right)^{-n},$$
where \(P_n(x)\) is a polynomial of degree \(n-1\). [6]
Assume that ( tan−1x ) =P (x)( 1+x2)−k , where degP (x)=k−1.
Answer
Marks
Guidance
dxk k k
B1
States inductive hypothesis. Must have
degP (x)=k−1.
k
dk+1( tan−1x )
'(x)( 1+x2)−k (x)( 1+x2)−k−1
=P −2kxP
Answer
Marks
Guidance
dxk+1 k k
M1 A1
Differentiates kth derivative using the product
rule.
= ( P '(x)( 1+x2) −2kxP (x) )( 1+x2)−k−1 so degP (x)=k
Answer
Marks
Guidance
k k k+1
A1
(x)( 1+x2)−k−1
Writes in the form P .
k+1
Answer
Marks
Guidance
So true when n=k+1. By induction, true for all positive integers n.
A1
Attempts to show degree of P (x) is at most k
k+1
(condone not showing coefficient of xk is non-
zero) and states conclusion.
6
Answer
Marks
Guidance
Question
Answer
Marks
Question 2:
2 | d ( tan−1x ) 1
= so true when n=1.
dx 1+x2 | B1 | Differentiates once.
dk
Assume that ( tan−1x ) =P (x)( 1+x2)−k , where degP (x)=k−1.
dxk k k | B1 | States inductive hypothesis. Must have
degP (x)=k−1.
k
dk+1( tan−1x )
'(x)( 1+x2)−k (x)( 1+x2)−k−1
=P −2kxP
dxk+1 k k | M1 A1 | Differentiates kth derivative using the product
rule.
= ( P '(x)( 1+x2) −2kxP (x) )( 1+x2)−k−1 so degP (x)=k
k k k+1 | A1 | (x)( 1+x2)−k−1
Writes in the form P .
k+1
So true when n=k+1. By induction, true for all positive integers n. | A1 | Attempts to show degree of P (x) is at most k
k+1
(condone not showing coefficient of xk is non-
zero) and states conclusion.
6
Question | Answer | Marks | Guidance
Prove by mathematical induction that, for all positive integers $n$,
$$\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(\tan^{-1}x\right) = P_n(x)\left(1+x^2\right)^{-n},$$
where $P_n(x)$ is a polynomial of degree $n-1$. [6]
\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q2 [6]}}