To determine the derivative of the functions from first principles, we use the definition:

April 3, 2026Mathematics

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April 3, 2026

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\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

To determine the derivative of the functions from first principles, we use the definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

a) $y = \sec x$ Here, $f(x) = \sec x = \frac{1}{\cos x}$ . $f'(x) = \lim_{h \to 0} \frac{\sec(x+h) - \sec x}{h}$ $f'(x) = \lim_{h \to 0} \frac{\frac{1}{\cos(x+h)} - \frac{1}{\cos x}}{h}$

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To determine the derivative of the functions from first principles, we use the definition: f'(x) = _h 0 (f(x+h) - f(x))/(h) a) y = x Here, f(x) = x = (1)/( x).

Asked on April 3, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on April 3, 2026

To determine the derivative of the functions from first principles, we use the definition: f'(x) = _h 0 (f(x+h) - f(x))/(h) a) y = x Here, f(x) = x = (1)/( x). f'(x) = _h 0 ((x+h) - x)/(h) f'(x) = _h 0 (1)/((x+h)) - (1)/( x)h f'(x) = _