Differentiation Rules

1. The Sum Rule

In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist.[1]

Given: h(x)=f(x)+g(x)

Proofs: h′(x)=f′(x)+g′(x)

h′(x)=limΔx→0h(x+Δx)−h(x)Δx=limΔx→0f(x+Δx)+g(x+Δx)−f(x)−g(x)Δx=limΔx→0f(x+Δx)−f(x)+g(x+Δx)−g(x)Δx=limΔx→0f(x+Δx)−f(x)Δx+g(x+Δx)−g(x)Δx=f′(x)+g′(x)

2. The Product Rule

In calculus, the product rule is a formula used to find the derivatives of products of two or more functions.[2]

Given: h(x)=f(x)⋅g(x)

Proofs: h′(x)=f′(x)g(x)+f(x)g′(x)

h′(x)=limΔx→0h(x+Δx)−h(x)Δx=limΔx→0f(x+Δx)⋅g(x+Δx)−f(x)⋅g(x)Δx=limΔx→0f(x+Δx)⋅g(x+Δx)−[f(x)⋅g(x+Δx)+f(x)⋅g(x+Δx)]−f(x)⋅g(x)Δx=limΔx→0[f(x+Δx)−f(x)]⋅g(x+Δx)+f(x)⋅[g(x+Δx)−g(x)]Δx=limΔx→0f(x+Δx)−f(x)Δx⋅limΔx→0g(x+Δx)+limΔx→0f(x+Δx)⋅limΔx→0g(x+Δx)−g(x)Δx=f′(x)g(x)+f(x)g′(x)

3. The Quotient Rule

In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions.[3]

Given: h(x)=f(x)g(x)

Proofs: h′(x)=f′(x)g(x)−f(x)g′(x)g(x)2

h′(x)=limΔx→0h(x+Δx)−h(x)Δx=limΔx→0f(x+Δx)g(x+Δx)−f(x)g(x)Δx=limΔx→0f(x+Δx)g(x)−f(x)g(x+Δx)Δx⋅g(x)g(x+Δx)=limΔx→0f(x+Δx)g(x)−f(x)g(x+Δx)Δx⋅1g(x)g(x+Δx)=[limΔx→0f(x+Δx)−f(x)Δx⋅limΔx→0g(x)−limΔx→0f(x)⋅limΔx→0g(x+Δx)−g(x)Δx]⋅1g(x)2=f′(x)g(x)−f(x)g′(x)g(x)2

Reference

[1] Wikipedia-Sum rule in differentiation

[2] Wikipedia-Product rule

[3] Wikipedia-Quotient_rule