Question

1. if ( y = \tan x - cot x ), then ( dy/dx = )

a ( sec x csc x )
b ( sec x - csc x )
c ( sec x + csc x )
d ( sec^{2}x - csc^{2}x )
e ( sec^{2}x + csc^{2}x )

Explanation:

Step1: Recall derivative formulas

Recall that \(\frac{d}{dx}(\tan x)=\sec^{2}x\), \(\frac{d}{dx}(\cot x)=-\csc^{2}x\)

Step2: Differentiate \(y = \tan x-\cot x\)

Using the sum - difference rule \(\frac{d}{dx}(u - v)=\frac{du}{dx}-\frac{dv}{dx}\), where \(u=\tan x\) and \(v = \cot x\)

\(\frac{dy}{dx}=\frac{d}{dx}(\tan x)-\frac{d}{dx}(\cot x)\)

Substitute the derivative formulas:

\(\frac{dy}{dx}=\sec^{2}x-(-\csc^{2}x)=\sec^{2}x+\csc^{2}x\)

Answer:

E. \(\sec^{2}x+\csc^{2}x\)