Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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What is the derivative of the function u^n (where n is a constant)?

nu^(n-1) du/dx
n/u^n du/dx
(1/n) u^n-1 du/dx
u^(n+1) du/dx

The correct answer is: nu^(n-1) du/dx

The derivative of the function $ u^n $, where $ n $ is a constant, is derived using the power rule of differentiation. According to the power rule, if you have a function of the form $ f(x) = x^n $, its derivative is given by $ f'(x) = n \cdot x^{n-1} $. In the expression $ u^n $, $ u $ can be viewed as a function of $ x $ (i.e., $ u = u(x) $). When taking the derivative, we need to apply the chain rule in addition to the power rule. The chain rule states that if you have a composite function $ f(g(x)) $, the derivative is $ f'(g(x)) \cdot g'(x) $. Applying this to $ u^n $, we find: 1. Differentiate $ u^n $ using the power rule: The derivative is $ n \cdot u^{n-1} $. 2. Multiply by the derivative of $ u $ with respect to $ x $ (which is $ du/dx $). Putting this together gives the final result for the derivative: