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:[tex] \sec(x) \times \cos(x) = 1[/tex]Step 2:[tex] \frac{d}{dx} ( \sec(x) \times \cos(x) ) = 0[/tex]Step 3:[tex] \frac{d}{dx} (\sec(x)) \times \cos(x) - \sec(x) = 0[/tex]Step 4:[tex] \frac{d}{dx} \sec(x) = \frac{ \sec(x) \times \sin(x) }{ \cos(x) } = \sec(x) \times \tan(x) [/tex]A. step 1B. Step 2C. Step 3D. There is no error​

Accepted Solution

The error occurs in step 3. By the product rule, we have[tex]\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)[/tex][tex]=\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x\boxed{+\sec x\times(-\sin x)}[/tex][tex]=\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x\boxed{-\sec x\times\sin x}[/tex](i.e. there is a missing factor of [tex]\sin x[/tex])Then[tex]\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x-\sec x\times\sin x=0[/tex][tex]\implies\dfrac{\mathrm d}{\mathrm dx}(\sec x)\times\cos x=\sec x\times\sin x[/tex][tex]\implies\dfrac{\mathrm d}{\mathrm dx}(\sec x)=\dfrac{\sec x\times\sin x}{\cos x}[/tex][tex]\implies\dfrac{\mathrm d}{\mathrm dx}(\sec x)=\sec x\times\tan x[/tex]