G: Differentiation: Understand and use the derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change; sketching the gradient function for a given curve; second derivatives; differentiation from first principles for small positive integer powers of x and for sin x and cos x. Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves and points of inflection.

The idea

The derivative f'(x) gives the gradient of the tangent to y = f(x) at a general point (x, y). It is defined as the limit of [f(x+h) - f(x)]/h as h tends to 0. This limit is the gradient of the tangent, and it also represents an instantaneous rate of change, such as velocity or marginal cost. You can sketch the gradient function by noting where f is increasing (f' > 0), decreasing (f' < 0), and stationary (f' = 0). The second derivative f''(x) is the derivative of f'(x); it measures how the gradient changes. Where f'' > 0 the curve is convex (concave up), where f'' < 0 it is concave (concave down), and a point of inflection occurs where the concavity changes, often when f'' = 0 and changes sign. You must differentiate from first principles for small positive integer powers of x and for sin x and cos x.

What earns the marks

• Interpret the derivative as the gradient of the tangent and as an instantaneous rate of change in context.
• Use the limit definition to differentiate from first principles for functions such as x^n (small positive integer n), sin x, and cos x.
• Sketch the graph of the gradient function for a given curve, identifying where the derivative is positive, negative, or zero.
• Calculate second derivatives and interpret them as the rate of change of the gradient.
• Use the sign of the second derivative to determine convexity/concavity and identify points of inflection.

Common mistakes and how to put them right

• Confusing the derivative with the original function.

  Put it right: f'(x) gives the gradient, not the y-value; for example, if f(x) = x^2, then f'(x) = 2x, which is the gradient at x.

• Thinking that f''(x) = 0 always means a point of inflection.

  Put it right: A point of inflection requires a change in concavity, so check that f'' changes sign; for example, y = x^4 has f''(0) = 0 but no inflection.

• Incorrectly applying first principles by omitting the limit or making algebraic errors.

  Put it right: Write the full definition, expand carefully, and take the limit as h tends to 0.

Exam tips

• When asked to differentiate from first principles, always start with the definition and show the limit step clearly.
• For sketching the gradient function, mark key points such as stationary points and where the gradient is positive or negative.
• When using the second derivative to determine concavity, state the sign of f'' and interpret it in terms of the curve's shape.