Mastering Derivatives: Sum, Difference, Product, and Chain Rule Explained

1 Derivative of Sum, Difference, and Product

In the previous article we learned how to differentiate simple functions; now we combine those results to derive rules for more complex functions.

1.1 Derivative of a Sum

Using the limit definition of the derivative, we obtain the sum rule, which states that the derivative of a sum equals the sum of the derivatives.

1.2 Derivative of a Difference

The derivative of a difference follows analogously, giving the difference of the derivatives.

1.3 Derivative of a Product

Another important rule is the product rule. Starting from the definition, we rearrange the numerator and recognize that the limit of the third term is zero, leading to the formula \( (fg)' = f'g + fg' \).

2 Functional Composition

Given a function that maps a set of elements to another set, and a second function that maps the result to a third set, we can combine them into a new function whose domain is the original set and whose codomain is the final set. The composition is performed by applying the inner function first, then the outer function.

For example, composing the two functions \(f\) and \(g\) yields the composite function \(g\circ f\).

3 Chain Rule

For a composite function \(h(x)=g(f(x))\), the derivative is given by the chain rule: \(h'(x)=g'(f(x))\cdot f'(x)\). This follows directly from the definition and the product rule applied to the composition.