Explain the chain rule of differentiation

The chain rule can be used to find more complex derivatives.For example, in the case of: y = (5x + 2)⁵To find the derivative in the ordinary fashion, one would need to expand the brackets to:y=3125x⁵+6250x⁴+5000x³+2000x²+400x+32If you persevere to this point the risk of human error is huge, so clearly an easier method is needed.Enter the chain rule:dy/dx = dy/du * du/dxOne sets u = 5x + 2Now y = u⁵ and u = 5x + 2Differentiate y wrt u:dy/du = 5u⁴Differentiate u wrt x:du/dx = 5 Substitute u into dy/dudy/du = 5(5x+2)⁴Recall that dy/dx = dy/du * du/dx:dy/dx = 5 * 5(5x+2)⁴dy/dx = 25(5x+2)⁴The chain rule can be expanded with as many terms as possible, and this is useful when considering real life rates of change:dy/dx = dy/du₁ * du₁/du₂ * du₂/du₃ * ... * du_n-1/du_n * du_n/dx