Differentiation

You will be able to

• Calculate the gradient of the chord joining x and x + h, and simplify [f(x + h) − f(x)] ÷ h algebraically before letting h tend to 0.
• Deduce the gradient function of xⁿ (n rational) and of sums and multiples of such terms as the limit of the chord gradients, using f′(x) and dy/dx.
• Explain why h = 0 cannot be substituted straight into the chord formula (0 ÷ 0) and why a chord is never the tangent for h ≠ 0.

You will be able to

• Locate the stationary points of a cubic, of x + k/x and of 2x^(3/2) − kx by solving dy/dx = 0, and give their coordinates exactly.
• Determine whether each stationary point is a maximum, a minimum or neither, using the sign of d²y/dx² or the sign of dy/dx either side, and explain why d²y/dx² = 0 does not decide.
• State the intervals on which y is increasing and decreasing, and use the stationary points to sketch the curve.