How to Find the Derivative of 1/x from First Principles
How to Find the Derivative of 1/x from First Principles

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Last updated at May 29, 2023 by Teachoo

Example 15 (Method 1) Find the derivative of f(x) = x + 1x f(x) = 𝑥 + 1 𝑥 = 𝑥𝑥 + 1𝑥 = 1 + 1𝑥 = 1 + (x) –1 Now, f’ (x) = (1 + (x) –1)’ = 0 + ( –1) x –1 – 1 = 0 – x –2 = – x –2 = −𝟏 𝒙𝟐 Example 15 (Method 2) Find the derivative of f(x) = x + 1x Given f(x) = 𝑥 + 1 𝑥 Let u = x + 1 & v = x So, f(x) = 𝑢𝑣 Now, f’(x) = 𝑢𝑣′ Using quotient rule f’(x) = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 Finding u’ and v’ separately Finding u’ & v’ u = x + 1 u’ = 1 + 0 = 1 v = x v’ = 1 Now, f’(x) = 𝑢𝑣′ = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 = 1 𝑥 − 𝑥 + 1 1 𝑥2 = 𝑥 − 𝑥 + 1 𝑥2 = 𝟏 𝒙𝟐

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