Exercise 5.4 · Q1: Differentiate $\dfrac{e^x}{\sin x}$

Basics you need

Solution

1. Where is the function defined?

   The numerator $e^x$ is defined for every real $x$⁠. The only danger is the denominator: we cannot divide by $0$⁠, so we need $\sin x \ne 0$⁠, that is

   $$x \ne n\pi, \qquad n \in \Z.$$

   So the domain is all real numbers except $0, \pm\pi, \pm 2\pi, \dots$

   A derivative can only exist where the function itself exists. Writing the domain down first also tells us in advance which condition the final formula will carry: the same $\sin x$ will reappear, squared, in its denominator.
   $\sin x = 0$ exactly at the integer multiples of $\pi$⁠. Basics: trigonometric identities

2. Name the top and the bottom, and differentiate each

   $$\begin{aligned} u &= e^x, & u' &= e^x, \\ v &= \sin x, & v' &= \cos x. \end{aligned}$$
   The quotient rule is built from four ingredients: $u$⁠, $v$ and their derivatives. Listing them separately before substituting avoids the most common error, mixing up which derivative goes where. $u' = e^x$ is Theorem 5 (1); $v' = \cos x$ is from Table 5.3.
   Theorem 5, Section 5.4. Lesson: exponential and logarithmic functions

3. Apply the quotient rule

   For $x \ne n\pi$ (so that $v = \sin x \ne 0$⁠):

   $$\begin{aligned} \frac{dy}{dx} &= \frac{u'v - uv'}{v^2} \\ &= \frac{e^x\cdot\sin x - e^x\cdot\cos x}{\sin^2 x}. \end{aligned}$$
   The rule reads "bottom times derivative of top, minus top times derivative of bottom, all over bottom squared". Here the "derivative of top" is $e^x$ again, which is why $e^x$ appears in both terms of the numerator.
   Quotient rule, rule (3) of Section 5.3, valid wherever the denominator is not zero. Basics: derivative rules

4. Factor, and say where the result holds

   Both terms of the numerator contain $e^x$⁠, so take it out:

   $$\frac{dy}{dx} = \frac{e^x(\sin x - \cos x)}{\sin^2 x},$$

   for $x \ne n\pi$⁠, $n \in \Z$⁠. The denominator $\sin^2 x$ is positive at every point of the domain, so the formula gives a real number at every $x$ where the function is defined.

   Factoring makes the sign of the derivative easy to read. Since $e^x > 0$ and $\sin^2 x > 0$⁠, the derivative has the same sign as $\sin x - \cos x$⁠. For example it is zero where $\tan x = 1$ (such as $x = \frac{\pi}{4}$⁠), where the graph has a horizontal tangent.
   $e^x > 0$ for every $x$⁠: the range of an exponential function is the positive reals (Section 5.4). Basics: exponentials and logarithms

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