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प्रश्न
Find the derivatives of the following functions using first principle.
f(x) = – x2 + 2
उत्तर
f(x + Δx) = – (x + Δx)2 + 2
f(x + Δx) – f(x) = – [x2 + 2x Δx + (Δx)2] + 2 – [– x2 + 2]
`(f(x + Deltax) - f(x))/(Deltax) = (- x^2 + 2xDeltax - (Deltax)^2 + 2 + x^2 - 2)/(Deltax)`
`(f(x + Deltax) - f(x))/(Deltax) = (- 2xDeltax - (Deltax)^2)/(Deltax)`
`(f(x + Deltax) - f(x))/(Deltax) = (-2xDeltax)/(Deltax) - (Deltax)^2/(Deltax)`
`(f(x + Deltax) - f(x))/(Deltax) = - 2x - Deltax`
`lim_(Deltax -> 0) (f(x + Deltax) - f(x))/(Deltax) = lim_(Deltax -> 0) (- 2x) - lim_(Deltax -> 0) Deltax`
`f"'"(x) = - 2x - 0`
`f"'"(x) = - 2x`
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