Finding the Derivative of a Function

📌 CONCEPT: To find the derivative of a function that cannot be differentiated using the power rule, we use the quotient rule of differentiation, which states that the derivative of a quotient of two functions is the quotient of their derivatives, provided the denominator is non-zero.

📐 RULE / FORMULA: According to the quotient rule, if f(x) = g(x)/h(x), then f'(x) = (h(x)g'(x) - g(x)h'(x)) / h(x)^2.

💡 WORKED EXAMPLE: Let's find the derivative of f(x) = (2x^2 - 5) / (x^2 + 1) at x = 2. We first identify g(x) = 2x^2 - 5 and h(x) = x^2 + 1. Then, g'(x) = 4x and h'(x) = 2x. Substituting these values into the quotient rule formula, we get f'(x) = ((x^2 + 1)(4x) - (2x^2 - 5)(2x)) / (x^2 + 1)^2. Simplifying, we get f'(x) = (4x^3 + 4x + 4x^3 - 10x) / (x^2 + 1)^2. Finally, we evaluate f'(2) by substituting x = 2 into the derivative function.

⚠️ COMMON MISTAKE: Students often forget to check if the denominator is non-zero before applying the quotient rule, which can lead to incorrect results.