Edexcel A-Level Mathematics A2 Fp2 6 1 Third and Higher Order Derivatives Questions

Only two direct questions exist, so focus on the repeated-derivative chain: keep each derivative visible, spot the pattern and match the required constant or coefficient.

Exam points

Differentiate e^x sin x repeatedly to prove y^(6)=k y'' and determine k.
Differentiate tan²x up to d³y/dx³, then identify the integer coefficients.

[Maximum number: 4]

Given that $y=e^x\sin x$

show that

\frac{\mathrm{d}^{6} y}{\mathrm{~d} x^{6}}=k \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}

where k is a constant to be determined.

y=e^x\sin x,

y'=e^x\sin x+e^x\cos x,

y''=2e^x\cos x,

y'''=2e^x\cos x-2e^x\sin x,

y^{(4)}=-4e^x\sin x=-4y.

Hence

y^{(6)}=-4y''

so k=-4.

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Edexcel A-Level Mathematics A2 Fp2 6 1 Third and Higher Order Derivatives Questions question 1