First principles: other functions

A question is this type if and only if it asks students to differentiate from first principles a function that is not polynomial, trigonometric, or reciprocal (e.g., square root, exponential, or composite functions).

1 questions · Standard +0.3

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Pre-U Pre-U 9794/2 2016 June Q11
12 marks Standard +0.3
The function f is defined by \(f(x) = \sqrt{x}, x > 0\).
  1. Use differentiation from first principles to find an expression for \(f'(x)\). [5]
The lines \(l_1\) and \(l_2\) are the tangents to the curve \(y = f(x)\) at the points \(A\) and \(B\) where \(x = a\) and \(x = b\) respectively, \(a \neq b\).
    1. Show that the tangents intersect at the point \(\left(\sqrt{ab}, \frac{1}{2}(\sqrt{a} + \sqrt{b})\right)\). [5]
    2. Given that \(l_1\) and \(l_2\) intersect at a point with integer coordinates, write down a possible pair of values for \(a\) and \(b\). [2]