Challenging +1.2 This is a structured induction proof on derivatives of hyperbolic functions. While it requires familiarity with hyperbolic function derivatives and careful algebraic manipulation through the inductive step, the framework is standard and the pattern is given explicitly. It's harder than routine A-level questions due to the induction on derivatives and hyperbolic functions (Further Maths content), but follows a predictable structure without requiring novel insight.
6 Let \(\mathrm { y } = \mathrm { x } \cosh \mathrm { x }\).
Prove by induction that, for all integers \(n \geqslant 1 , \frac { d ^ { 2 n - 1 } y } { d x ^ { 2 n - 1 } } = x \sinh x + ( 2 n - 1 ) \cosh x\).
Evidence of correct differentiation using product rule and a
substitution n = 1 into RHS
Differentiate using the product rule
Differentiate second time using the product rule
Must be equated to correct derivative form
Dependent on previous A1 and the B1
Question 6:
6 | Base case:
dy
y = xcoshx⇒ =coshx+xsinhx
dx
=(2×1−1)coshx+xsinhx
So true for n = 1
Assume result holds for n = k
d2k−1y
=xsinhx+(2k−1)coshx
dx2k−1
d2ky
⇒ =sinhx+xcoshx+(2k−1)sinhx
dx2k
d2k+1y
⇒ =coshx+coshx+xsinhx+(2k−1)coshx
dx2k+1
=xsinhx+(2k+1)coshx
d2(k+1)−1y
i.e. =xsinhx+(2(k+1)−1)coshx
dx2(k+1)−1
So if true for n = k then also true for n = k +
1
But it is true for n= 1 and so is true generally | B1
M1*
M1dep
M1dep
A1*
A1dep
[6] | 2.5
2.1
1.1
3.1a
2.2a
2.4 | Evidence of correct differentiation using product rule and a
substitution n = 1 into RHS
Differentiate using the product rule
Differentiate second time using the product rule
Must be equated to correct derivative form
Dependent on previous A1 and the B1
6 Let $\mathrm { y } = \mathrm { x } \cosh \mathrm { x }$.\\
Prove by induction that, for all integers $n \geqslant 1 , \frac { d ^ { 2 n - 1 } y } { d x ^ { 2 n - 1 } } = x \sinh x + ( 2 n - 1 ) \cosh x$.
\hfill \mbox{\textit{OCR Further Pure Core 1 2022 Q6 [6]}}