Summary of differentiation rules

We can summarise the differentiation rules we have found as follows. They can be expressed in both functional and Leibniz notation. First, the linearity of differentiation.

Linearity of differentiation
Functional notation Leibniz notation
Constant multiple \(\dfrac{d}{dx} \bigl[cf(x)\bigr]\) = \(cf'(x)\) \(\dfrac{d}{dx}\bigl(cf\bigr)\) = \(c\,\dfrac{df}{dx}\)
Sum \(\dfrac{d}{dx} \bigl[f(x) + g(x)\bigr]\) = \(f'(x) + g'(x)\) \(\dfrac{d}{dx} \bigl(f + g\bigr)\) = \(\dfrac{df}{dx} + \dfrac{dg}{dx}\)
Difference \(\dfrac{d}{dx} \bigl[f(x) - g(x)\bigr]\) = \(f'(x) - g'(x)\) \(\dfrac{d}{dx} \bigl(f - g\bigr)\) = \(\dfrac{df}{dx} - \dfrac{dg}{dx}\)

We also have the product, quotient and chain rules. In Leibniz notation, these rules are often written with \(u,v\) rather than \(f,g\).

Product, quotient and chain rules
Functional notation Leibniz notation
Product \(\dfrac{d}{dx} \bigl[f(x)\,g(x)\bigr]\) = \(f(x)\,g'(x) + g(x)\,f'(x)\) \(\dfrac{d}{dx} \bigl(uv\bigr)\) = \(u\,\dfrac{dv}{dx} + v\,\dfrac{du}{dx}\)
Quotient \(\dfrac{d}{dx} \Biggl[\dfrac{f(x)}{g(x)}\Biggr]\) = \(\dfrac{g(x)\,f'(x) - f(x)\,g'(x)}{(g(x))^2}\) \(\dfrac{d}{dx} \Biggl(\dfrac{u}{v}\Biggr)\) = \(\dfrac{v\,\dfrac{du}{dx} - u\,\dfrac{dv}{dx}}{v^2}\)
Chain \(\dfrac{d}{dx} \bigl[f(g(x))\bigr]\) = \(f'(g(x))\,g'(x)\) \(\dfrac{dy}{dx}\) = \(\dfrac{dy}{du}\,\dfrac{du}{dx}\)

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