Quotient Rule Practice Find the derivatives of the following rational functions. a) x2 x+1 b) x4 + 1 x2 c) sin(x) x Solution a) x2 x+1 The quotient rule tells us that if u(x) and v(x) are differentiable functions, and v(x) is non-zero, then: 0 u(x) u0 (x) v(x) − u(x) v 0 (x) = v(x) (v(x))2 In this problem u = x2 and v = x + 1, so u0 = 2x and v 0 = 1. Applying the quotient rule, we see that: 2 0 x 2x · (x + 1) − x2 · 1 = x+1 (x + 1)2 2x2 + 2x − x2 = (x + 1)2 2 x + 2x = . (x + 1)2 b) x4 + 1 x2 Here u(x) = x4 + 1, u0 (x) = 4x3 , v(x) = x2 and v 0 (x) = 2x. The quotient rule tells us that: 4 0 x +1 4x3 · x2 − (x4 + 1) · 2x = 2 x (x2 )2 4x5 − 2x5 − 2x = x4 4 2x − 2 = . x3 c) sin(x) x The derivative of sin(x) is cos(x) and the derivative of x is 1, so the quotient rule tells us that: sin(x) x 0 = = (cos(x)) · x − (sin(x)) · 1 x2 x cos(x) − sin(x) . x2 When we learn to evaluate this expression at x = 0, it will tell us that the slope of the graph of sinc(x) = sinx x is 0 when x = 0.

Quotient Rule Practice

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