1.07g Differentiation from first principles: for small positive integer powers of x

A curve has equation $$y = 4x^2 - 5x$$ The curve passes through the point \(P(2, 6)\) Use differentiation from first principles to find the value of the gradient of the curve at \(P\). [4]

Given the function \(f(x) = x - x^2\), defined for all real values of \(x\),

1. Show that \(f'(x) = 1 - 2x\) by differentiating \(f(x)\) from first principles. [4]
2. Find the maximum value of \(f(x)\). [1]
3. Explain why \(f^{-1}(x)\) does not exist. [1]

Given the function \(f(x) = 3x^3 - 7x - 1\), defined for all real values of \(x\), prove from first principles that \(f'(x) = 9x^2 - 7\). [4]

Differentiate \(f(x) = x^4\) from first principles. [5]

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. 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]