Question

multiple choice 2 points
what is the derivative of ( 9 cos ( x ) - ln ( x ) )?
( - 9 sin ( x ) - \frac { 1 } { x } )
( 9 sin ( x ) - \frac { 1 } { x } )
( - sin ( 9 x ) - \frac { 1 } { x } )
( - 9 sin ( x ) - x ln ( x ) )
( sin ( 9 x ) - \frac { 1 } { x } )
( 9 sin ( x ) - x ln ( x ) )
clear my selection

Explanation:

Step1: Differentiate \(9\cos(x)\)

The derivative of \(\cos(x)\) is \(-\sin(x)\). Using the constant multiple rule \((cf(x))^\prime = cf^\prime(x)\) (where \(c = 9\) and \(f(x)=\cos(x)\)), the derivative of \(9\cos(x)\) is \(9\times(-\sin(x))=-9\sin(x)\).

Step2: Differentiate \(-\ln(x)\)

The derivative of \(\ln(x)\) is \(\frac{1}{x}\). Using the rule \((-f(x))^\prime=-f^\prime(x)\) (where \(f(x)=\ln(x)\)), the derivative of \(-\ln(x)\) is \(-\frac{1}{x}\).

Step3: Combine the derivatives

Using the sum - difference rule \((u\pm v)^\prime=u^\prime\pm v^\prime\) (where \(u = 9\cos(x)\) and \(v=\ln(x)\)), the derivative of \(y = 9\cos(x)-\ln(x)\) is \((9\cos(x))^\prime-(\ln(x))^\prime=-9\sin(x)-\frac{1}{x}\).

Answer:

\(-9\sin(x)-\frac{1}{x}\) (the first option)