Derivative of tan x

Apply the quotient rule to $\tan x=\dfrac{\sin x}{\cos x}$. The numerator becomes $\cos^2 x+\sin^2 x=1$.

You can memorize the result, but derive it by hand once so it sticks.

Examples

1. 01Differentiate:$\displaystyle \tan x$AnswerHide

   Answer$\displaystyle \frac{1}{\cos^{2}x}$

   Steps

   1. Quotient rule on $\tan x=\dfrac{\sin x}{\cos x}$: the numerator collapses to $\cos^{2}x+\sin^{2}x$
   2. $\displaystyle \frac{\cos^{2}x+\sin^{2}x}{\cos^{2}x}$$\displaystyle = \frac{1}{\cos^{2}x}$
   3. The denominator is the original $\cos^{2}x$; not $\sin^{2}x$
2. 02What is the value?$\displaystyle \cos^{2}x+\sin^{2}x$AnswerHide

   Answer$\displaystyle 1$

   Steps

   1. This holds for every x
   2. $\displaystyle \sin^{2}x+\cos^{2}x$$\displaystyle = 1$
   3. At $x=0$ for instance, $\sin 0=0$ and $\cos 0=1$, and $0+1=1$
3. 03What is the value at x=0?$\displaystyle (\tan x)'=\frac{1}{\cos^{2}x}$AnswerHide

   Answer$\displaystyle 1$

   Steps

   1. Put in $\cos 0 = 1$
   2. $\displaystyle \frac{1}{1^{2}}$$\displaystyle = 1$
   3. Slope 1; the graph of $\tan x$ leaves the origin at 45 degrees
4. 04What goes in □?$\displaystyle \tan x = \frac{\sin x}{\square}$AnswerHide

   Answer$\displaystyle \cos x$

   Steps

   1. This is what $\tan$ means
   2. $\displaystyle \tan x$$\displaystyle = \frac{\sin x}{\cos x}$
   3. At $x=\dfrac{\pi}{4}$, for instance, $\sin$ and $\cos$ are equal so the quotient is 1; matching $\tan\dfrac{\pi}{4}=1$
5. 05Differentiate$\displaystyle \frac{d}{dx} \tan x$AnswerHide

   Answer$\displaystyle \frac{1}{\cos^2 x}$
6. 06Expand:$\displaystyle \cos x\cos x-\sin x(-\sin x)$AnswerHide

   Answer$\displaystyle \cos^{2}x+\sin^{2}x$

FAQ