Properties of derivatives

The derivative of the sum or a difference of two functions is the sum or the difference (respectively) of their derivatives. Given functions \(f(x)=u(x)\pm v(x)\)

Lagrange notation: \(f'=u'\pm v'\)

Leibniz notation: \(\dfrac{df}{dx}=\dfrac{du}{dx}\pm\dfrac{dv}{dx}\)

Note: It is as if the derivative is distributed over addition and subtraction

The product rule is applied when we want to get the derivative of the product of two functions. Given \(f(x)=u(x)\cdot v(x)\)

Lagrange notation: \(f'=u'\cdot v+u\cdot v'\)

Leibniz notation: \(\dfrac{df}{dx}=\dfrac{du}{dx}\cdot v+u\cdot\dfrac{dv}{dx}\)

Note: the rule is a sum. Take the derivative of the first function first and the second function second. This helps to memorize the pattern as it bears some similarity to the quotient rule.

The extended product rule applies to the product of three (or more) functions. Given \(f(x)=u_1(x)\cdot u_2(x)\cdot u_3(x)\), the derivative is \[f'=u'_1\cdot u_2\cdot u_3+u_1\cdot u'_2\cdot u_3+u_1\cdot u_2\cdot u'_3\]

The quotient rule is applied when we want to get the derivative of the quotient of two functions. Given \(f(x)=\dfrac{u(x)}{v(x)}\)

Lagrange notation: \(f'=\dfrac{u'\cdot v-u\cdot v'}{v^2}\)

Leibniz notation: \(\dfrac{df}{dx}=\dfrac{\tfrac{du}{dx}\cdot v-u\cdot\tfrac{dv}{dx}}{v^2}\)

Note: the rule is a quotient, and its numerator is a difference. Take the derivative of the first function first and the second function second. This helps to memorize the pattern as it bears some similarity to the quotient rule.

The chain rule is applied when we have a composite function; i.e., a function of a function or a function within a function. The derivative is the product of their respective derivatives. Given \(f(x)=u\left[v(x)\right]\)

Lagrange notation: \(f'(x)=u'\left[v(x)\right]\cdot v'(x)\)

Leibniz notation: \(\dfrac{df}{dx}=\dfrac{du}{dv}\cdot \dfrac{dv}{dx}\)

Note: The argument of \(u'\) is the same as the argument of \(u\). This is a major difference from the other rules, where the derivative is \(x\) for all derivatives.

Note: In Leibniz notation, it is as if the ‘denominator’ of one derivative cancels out with the ‘numerator’ of the next.

If we have more than two nested functions, we take their derivatives from the inside out or the outside in. For example, given \(f(x)=u\left(v\left[w(x)\right]\right)\), \[f'(x)=u'\left(v\left[w(x)\right]\right)\cdot v'\left[w(x)\right]\cdot w'(x)\]