Final answer:
The derivative of the function f(x) = x^n + x^(n-1) is found by applying the power rule to each term separately. The correct derivative is f'(x) = n*x^(n-1) + (n-1)*x^(n-2), which differs from the expression provided by the student due to a likely typo or misunderstanding.
Step-by-step explanation:
Show Derivative of Polynomial Function
To find the derivative of the function f(x) = x^n + x^(n-1), we use the power rule of differentiation. The power rule states that if f(x) = x^m, then f'(x) = m*x^(m-1). Applying this to each term of the function separately, we have:
- For the first term x^n, the derivative is n*x^(n-1).
- For the second term x^(n-1), the derivative is (n-1)*x^((n-1)-1), which simplifies to (n-1)*x^(n-2).
Adding these two derivatives together, we get the overall derivative of the function: f'(x) = n*x^(n-1) + (n-1)*x^(n-2). However, the expression provided in the student's question f'(x) = nx + n - 1/x^(2-n) does not match the correct derivative and may be due to a typo or misunderstanding.
Therefore, to correct the student's conception, we express that the derivative is a sum of each term's derivative, not a single term as mistakenly given in the question.