Quick questions on Product, quotient and chain rules explained: O-Level A-Maths

For a quotient $y = \dfrac{u}{v}$:

Real functions mix the rules: a product whose factor is a composite, or a quotient whose parts are products. Identify the outermost structure first (is the whole thing a product, a quotient, or a composite?), then apply the chain rule wherever a composite appears inside.

A quick test of structure decides the rule. If the function is two things multiplied, use the product rule; if it is one thing divided by another, use the quotient rule; if it is a function inside another function (a power of a bracket, an exponential of an expression), use the chain rule. Many expressions need more than one rule, applied from the outside in.

The numerator is $u'v - uv'$ (top derivative first); reversing it flips the sign of the answer.

Differentiate $y = (x^2 + 1)^4$. [2 marks]

Differentiate $y = x\ln x$. [2 marks]

Differentiate $y = \dfrac{\sin x}{x}$. [3 marks]