Question

Below is an attempt to derive the derivative of sec(x) using product rule, where x is in the domain of secx. In which step, if any, does an error first appear?
Step 1:
$\sec(x) \times \cos(x) = 1$
Step 2:
$\frac{d}{dx} ( \sec(x) \times \cos(x) ) = 0$
Step 3:
$\frac{d}{dx} (\sec(x)) \times \cos(x) - \sec(x) = 0$
Step 4:
$\frac{d}{dx} \sec(x) = \frac{ \sec(x) \times \sin(x) }{ \cos(x) } = \sec(x) \times \tan(x)$

A. step 1
B. Step 2
C. Step 3
D. There is no error
​

Answer 1

The error occurs in step 3. By the product rule, we have

$\dfrac{\mathrm d}{\mathrm dx}(\sec x\times\cos x)=\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x+\sec x\times\dfrac{\mathrm d}{\mathrm dx}(\cos x)$

$=\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x\boxed{+\sec x\times(-\sin x)}$

$=\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x\boxed{-\sec x\times\sin x}$

(i.e. there is a missing factor of $\sin x$)

Then

$\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x-\sec x\times\sin x=0$

$\implies\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x=\sec x\times\sin x$

$\implies\dfrac{\mathrm d}{\mathrm dx}(\sec x)=\dfrac{\sec x\times\sin x}{\cos x}$

$\implies\dfrac{\mathrm d}{\mathrm dx}(\sec x)=\sec x\times\tan x$