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Limits and Derivatives · JEE Preparation Mathematics

Using the definition of derivative, find the derivative of f(x)=frac{1}{x} at x=2.

Practice question from Limits and Derivatives (JEE Preparation Mathematics), with the correct answer and a step-by-step solution.

medium · Subjective · Q-457DDA

Using the definition of derivative, find the derivative of f(x)=1xf(x)=\frac{1}{x} at x=2x=2.

Answer: By definition, f′(2)=lim⁡h→0f(2+h)−f(2)h=lim⁡h→012+h−12hf'(2)=\lim_{h\to 0}\frac{f(2+h)-f(2)}{h}=\lim_{h\to 0}\frac{\frac{1}{2+h}-\frac{1}{2}}{h}. Simplifying gives f′(2)=lim⁡h→0−12(2+h)=−14f'(2)=\lim_{h\to 0}\frac{-1}{2(2+h)}=-\frac{1}{4}. Hence, the derivative of f(x)=1/xf(x)=1/x at x=2x=2 is −1/4-1/4.

Solution

Apply the derivative definition: f′(2)=lim⁡h→0f(2+h)−f(2)hf'(2)=\lim_{h\to 0}\frac{f(2+h)-f(2)}{h}. Simplification yields lim⁡h→0−12(2+h)=−14\lim_{h\to 0}\frac{-1}{2(2+h)}=-\frac{1}{4}.

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