Further differentiation and Maclaurin series

Learning objectives

Differentiate inverse trigonometric and inverse hyperbolic functions, quoting results such as d/dx(arctan x) = 1/(1 + x²), d/dx(arcsin x) = 1/√(1 − x²) and d/dx(arsinh x) = 1/√(1 + x²)

Find the second derivative d²y/dx² of a relation defined implicitly, and of a curve given parametrically using d²y/dx² = (d/dt(dy/dx))/(dx/dt) rather than by dividing second derivatives

Derive a Maclaurin series by repeated differentiation, using f(x) = f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + ... , to obtain the first few terms (derivation of a general term is not required)

Obtain a Maclaurin expansion by repeatedly differentiating a relation satisfied by the function, such as (1 + x²)y′ = y, and evaluating the derivatives at x = 0 from the resulting recurrence

Recall the standard expansions of eˣ, sin x, cos x, ln(1 + x) and (1 + x)ⁿ together with the values of x for which each is valid, and combine or substitute into them to expand composite functions

Apply a Maclaurin series to evaluate a limit of indeterminate form or to approximate a value, and comment on how the accuracy depends on the number of terms and the size of x