Examples of using the chain rule, product rule and quotient rule for differentiation
f(x) = cos^3 (2x)
f(x) = 2x(5x^2 + 3)^4
Use product rule and chain rule
2x(4(5x^2 + 3)^3)(10x)
f(x) = 6x^2/(3x^2 + 4)^3
[(3x^2 + 4)^3(12x) - (6x^2)(3(3x^2 + 4)^2(6x))]/(3x^2 + 4)^6
Product rule is first term times derivative of the second plus second term times derivative of the first.
Quotient rule is denominator times derivative of numerator times numerator times derivative of denominator, all divided by denominator squared.