Find the derivative of x2 - 2 at x = 10 from first principle.

1.	Find the derivative of x2 - 2 at x = 10 from first principle.
Answer» Let, f(x) = x² − 2 ∴ f' (x) = \(\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}\) = \(\lim\limits_{h \to 0}\frac{[(x+h)^2-2]-(x^2-2)}{h}\) = \(\lim\limits_{h \to 0}\frac{x^2+2xh+h^2-2-x^2+2}{h}\) = \(\lim\limits_{h \to 0}\frac{2xh+h^2}{h}\) = \(\lim\limits_{h \to 0}(2x+h)\) = 2x + 0 = 2x ∴ At x = 10, f'(x) = 2 ∙10 = 20.

Answer»

Let,

f(x) = x² − 2

∴ f' (x) = \(\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}\)

= \(\lim\limits_{h \to 0}\frac{[(x+h)^2-2]-(x^2-2)}{h}\)

= \(\lim\limits_{h \to 0}\frac{x^2+2xh+h^2-2-x^2+2}{h}\)

= \(\lim\limits_{h \to 0}\frac{2xh+h^2}{h}\)

= \(\lim\limits_{h \to 0}(2x+h)\)

= 2x + 0

= 2x

∴ At x = 10,

f'(x) = 2 ∙10

= 20.

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