Differentiation formulae & properties

We start this section with some basic differentiation formulae.

The derivative of a constant is zero: If $f(x)=c$ where $c$ is a constant then $f^{\prime}(x)=0$ .
The power rule: If $f(x)=x^{n}$ where $n$ is a number and $x$ is a variable then:

f^{\prime}(x)=n x^{n-1} .

Differentiation properties

I. Multiplicative constant rule

\frac{d}{d x}[c f(x)]=c \frac{d f}{d x}

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This is a consequence of the multiplicative rule for limits: $\lim _{x \rightarrow a}[c f(x)]=c \lim _{x \rightarrow a} f(x)$ , where $c$ is a constant.

II. Sum/difference rule

\frac{d}{d x}[u(x) \pm v(x)]=u^{\prime}(x) \pm v^{\prime}(x)

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This result follows from another rule of limits: $\lim _{x \rightarrow a}[f(x) \pm g(x)]=\lim _{x \rightarrow a} f(x) \pm \lim _{x \rightarrow a} g(x)$ .

III. Product rule

\frac{d}{d x}[u(x) v(x)]=v(x) u^{\prime}(x)+u(x) v^{\prime}(x)

Starting from first principles and with a function $f(x)=u(x) v(x)$ , we have

\begin{aligned} \frac{f(x+\Delta x)-f(x)}{\Delta x} & =\frac{u(x+\Delta x) v(x+\Delta x)-u(x) v(x)}{\Delta x}, \\ & =\frac{u(x+\Delta x) v(x+\Delta x)-u(x) v(x+\Delta x)+u(x) v(x+\Delta x)-u(x) v(x)}{\Delta x}, \\ & =v(x+\Delta x) \frac{u(x+\Delta x)-u(x)}{\Delta x}+u(x) \frac{v(x+\Delta x)-v(x)}{\Delta x} . \end{aligned}

Now, as $\Delta x \rightarrow 0$ , the above approaches

v(x) u^{\prime}(x)+u(x) v^{\prime}(x) .

Note that the product rule can be extended to more than two functions.

IV. Quotient rule

\frac{d}{d x}\left[\frac{u(x)}{v(x)}\right]=\frac{u^{\prime}(x) v(x)-u(x) v^{\prime}(x)}{[v(x)]^{2}}

Similarly to the derivation of the product rule above, the derivative of the quotient is also proved using limits.

Definition of the derivative Chain rule