# Misc 1 (ii) - Chapter 12 Class 11 Limits and Derivatives

Last updated at April 16, 2024 by Teachoo

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Last updated at April 16, 2024 by Teachoo

Misc 1 Find the derivative of the following functions from first principle: (ii) (βπ₯)^(β1) Let f (x) = (βπ₯)^(β1) We need to find Derivative of f(x) i.e. fβ (x) We know that fβ(x) = limβ¬(hβ0) fβ‘γ(x + h) β f(x)γ/h Here, f (x) = (βπ₯)^(β1) So, f (x + h) = (β(π₯+β))^(β1) Putting values fβ(x) = limβ¬(hβ0)β‘γ(γ(β(π₯ + β)) γ^(β1) γβ (β π₯)γ^(β1))/βγ = limβ¬(hβ0)β‘γ((1/(β(π₯ + β))) β(1/(βπ₯)))/βγ = limβ¬(hβ0)β‘γ((β1)/(π₯ + β) + 1/π₯)/βγ = limβ¬(hβ0)β‘γ((β π₯ + π₯ + β)/(π₯ (π₯ + β) ))/βγ = limβ¬(hβ0)β‘γ(β π₯ + π₯ + β)/(βπ₯ (π₯ + β))γ = limβ¬(hβ0)β‘γβ/(βπ₯ (π₯ + β))γ = limβ¬(hβ0)β‘γ1/(π₯ (π₯ + β))γ Putting h = 0 = 1/(π₯ (π₯ + 0)) = π/ππ