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Differentiation (Metric)

Learn, or revise, differentiation from first principles, techniques of differentiation and using differentiation to sketch curves and solve problems.

Principles

The gradient of a curve as the limit of the gradient of a secant or chord. This is an A-level topic, but sometimes a bit neglected at that level.

Gradient of a curve at a point as the limit of the gradient of a secant or chord; the idea of the derivative.

Techniques

Differentiating key functions, and sums, products, quotients and composites. This is an A-level topic, which Science and Engineering courses at Imperial will assume you know.

Differentiating functions of the form \(a\,f(x)+b\,g(x)\).

Differentiating functions of the form \(f(x)\,g(x)\).

Differentiating functions of the form \(f(x)/g(x)\).

Differentiating functions of the form \(f(g(x))\).

Putting the rules of differentiation to use.

Differentiating functions given implicitly, in equations of the form \(f(x,\,y)=0\).

The \(n\)th derivative of a product.

Key results to do with differentiation.

Applications

Using differentiation to find stationary points and points of inflexion on curves, and to solve problems involving velocity and acceleration. This is an A-level topic, which Science and Engineering courses at Imperial will assume you know.

The result \(f(x)=f(0)+x\,f'(0)+x^2/2!\,f''(0)+\dots\).

Points where a curve is locally horizontal; how to find and classify them.

Finding points where a curve changes its direction of curvature.