Has your daughter registered as self-employed with the tax office yet? If not, she should do so within three months of starting out. Take a look here for information on how to do this.
If your daughter buys a new cello, she has several choices as far as using the cost to reduce her tax is concerned, as follows:
1 She could claim a 100% write off for the full cost in the year of purchase against her income under the annual investment allowance rules. Alternatively, she could claim something between 0% and 100% of the available cost under the AIA rules depending on the level of her income so as not to waste personal allowances. In 2013/14, an individual does not pay tax on the first £9,440 of their income so if your daughter's profit was less than that, there would probably be no point in claiming any capital allowances. If that was the case, the cost of the cello would fall into a pool of expenditure to be used in future tax years. This can all be decided at the end of the first accounting period.
2 She could claim an annual writing down allowance on the cost of the cello which would drip feed the tax relief over several tax years which may be beneficial if her earnings increase in the first few years and she is paying a higher rate of tax. The rate of the allowance is 18% so in year one she would claim a deduction of £540 (£3,000 x 18%), in year two she would claim £443 (£2,460 x 18%), in year three the allowance would be £363 £2,017 x 18%) and so on until the cost is written off.
3 She could claim nothing until her income made it worthwhile doing so in terms of reducing her tax liability.
Take a look here for some useful notes on capital allowances.
I hope this helps but let me know if you have any further questions.