G: Differentiation: Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points, points of inflection. Identify where functions are increasing or decreasing.

The idea

Differentiation gives the gradient function. To find the gradient at a point, substitute the x-coordinate into dy/dx. The tangent at that point has equation y - y1 = m(x - x1), where m is the gradient. The normal is perpendicular, so its gradient is -1/m (provided m ≠ 0). Stationary points occur where dy/dx = 0; solve this equation. Classify them using the second derivative: if d²y/dx² > 0, it's a minimum; if < 0, a maximum; if = 0, further investigation is needed. Points of inflection occur where the concavity changes, often when d²y/dx² = 0 and changes sign. A function is increasing where dy/dx > 0 and decreasing where dy/dx < 0.

For example: For y = x³ - 3x, dy/dx = 3x² - 3; stationary points at x = ±1; d²y/dx² = 6x, so x=1 is a minimum and x=-1 a maximum.

What earns the marks

• Find the gradient of a curve at a given point by evaluating dy/dx at that point.
• Determine the equation of the tangent or normal to a curve at a given point.
• Locate stationary points by solving dy/dx = 0 and classify them using the second derivative or sign of dy/dx.
• Identify points of inflection where the concavity changes, often using d²y/dx² = 0 and sign change.
• Determine intervals where a function is increasing (dy/dx > 0) or decreasing (dy/dx < 0).

Common mistakes and how to put them right

• Using the wrong gradient for the normal.

  Put it right: If tangent gradient is m, normal gradient is -1/m.

• Assuming any point where dy/dx = 0 is a maximum or minimum without checking.

  Put it right: Use the second derivative test or check sign change of dy/dx.

• Confusing increasing/decreasing with positive/negative function values rather than gradient.

  Put it right: Increasing means dy/dx > 0, decreasing means dy/dx < 0.

Exam tips

• For tangents and normals, clearly state the point and gradient before forming the equation.
• When classifying stationary points, show your method: either second derivative or a sign table.
• For increasing/decreasing, solve the inequality dy/dx > 0 or < 0 and state the intervals clearly.