Throughout the theorems which follow we assume that the functions $$f(x)$$ and $$F(x)$$ have derivatives $$f'(x)$$ and $$F'(x)$$ for the values of $$x$$ considered.

(1) If $$\phi(x) = f(x) + F(x)$$, then $$\phi(x)$$ has a derivative $\phi'(x) = f'(x) + F'(x).$

(2) If $$\phi(x) = kf(x)$$, where $$k$$ is a constant, then $$\phi(x)$$ has a derivative $\phi'(x) = kf'(x).$

We leave it as an exercise to the reader to deduce these results from the general theorems stated in Ex. XXXV. 1.

(3) If $$\phi(x) = f(x)F(x)$$, then $$\phi(x)$$ has a derivative $\phi'(x) = f(x)F'(x) + f'(x)F(x).$

For \begin{aligned} \phi'(x) &= \lim\frac{f(x + h)F(x + h) – f(x)F(x)}{h}\\ &= \lim\left\{f(x + h)\frac{F(x + h) – F(x)}{h} + F(x)\frac{f(x + h) – f(x)}{h}\right\}\\ &=f(x)F'(x) + F(x)f'(x).\end{aligned}

(4) If $$\phi(x) = \dfrac{1}{f(x)}$$, then $$\phi(x)$$ has a derivative $\phi'(x) = -\frac{f'(x)}{\{f(x)\}^{2}}.$

In this theorem we of course suppose that $$f(x)$$ is not equal to zero for the particular value of $$x$$ under consideration. Then $\phi'(x) = \lim \frac{1}{h} \left\{\frac{f(x) – f(x + h)}{f(x + h)f(x)}\right\} = -\frac{f'(x)}{\{f(x)\}^{2}}.$

(5) If $$\phi(x) = \dfrac{f(x)}{F(x)}$$, then $$\phi(x)$$ has a derivative $\phi'(x) = \frac{f'(x)F(x) – f(x)F'(x)}{\{F(x)\}^{2}}.$

This follows at once from (3) and (4).

(6) If $$\phi(x) = F\{f(x)\}$$, then $$\phi(x)$$ has a derivative $\phi'(x) = F’\{f(x)\} f'(x).$

For let $f(x) = y,\quad f(x + h) = y + k.$ Then $$k \to 0$$ as $$h \to 0$$, and $$k/h \to f'(x)$$. And \begin{aligned} \phi'(x) & = \lim \frac{F\{f(x + h)\} – F\{f(x)\}}{h}\\ & = \lim \left\{\frac{F(y + k) – F(y)}{k}\right\} \times \lim \left(\frac{k}{h}\right)\\ & = F'(y)f'(x).\end{aligned}

This theorem includes (2) and (4) as special cases, as we see on taking $$F(x) = kx$$ or $$F(x) = 1/x$$. Another interesting special case is that in which $$f(x) = ax + b$$: the theorem then shows that the derivative of $$F(ax + b)$$ is $$aF'(ax + b)$$.

Our last theorem requires a few words of preliminary explanation. Suppose that $$x = \psi(y)$$, where $$\psi(y)$$ is continuous and steadily increasing (or decreasing), in the stricter sense of § 95, in a certain interval of values of $$y$$. Then we may write $$y = \phi(x)$$, where $$\phi$$ is the ‘inverse’ function (§ 109) of $$\psi$$.

(7) If $$y = \phi(x)$$, where $$\phi$$ is the inverse function of $$\psi$$, so that $$x = \psi(y)$$, and $$\psi(y)$$ has a derivative $$\psi'(y)$$ which is not equal to zero, then $$\phi(x)$$ has a derivative $\phi'(x) = \frac{1}{\psi'(y)}.$

For if $$\phi(x + h) = y + k$$, then $$k \to 0$$ as $$h \to 0$$, and $\phi'(x) = \lim_{h \to 0} \frac{\phi(x + h) – \phi(x)}{(x + h) – x} = \lim_{k \to 0} \frac{(y + k) – y}{\psi(y + k) – \psi(y)} = \frac{1}{\psi'(y)}.$ The last function may now be expressed in terms of $$x$$ by means of the relation $$y = \phi(x)$$, so that $$\phi'(x)$$ is the reciprocal of $$\psi’\{\phi(x)\}$$. This theorem enables us to differentiate any function if we know the derivative of the inverse function.