multiple choice 2 points\nwhat is the derivative of ( 9 cos ( x ) - ln ( x ) )?\n( - 9 sin ( x ) - \frac { 1…

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

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

Answer

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)