What is the derivative of ln(xcos(x)) ?

The derivative of ln(xcos(x)) is (cos(x))/x-sin(x)ln(x)

What is the first derivative of ln(xcos(x)) ?

The first derivative of ln(xcos(x)) is (cos(x))/x-sin(x)ln(x)

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\frac{d}{d x} (ln (x) cos (x))

\frac{cos ( x )}{x} - sin (x) ln (x)

Solution steps

\frac{d}{d x} (ln (x) cos (x))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = ln (x), g = cos (x)

= \frac{d}{d x} (ln (x)) cos (x) + \frac{d}{d x} (cos (x)) ln (x)

\frac{d}{d x} (ln (x)) = \frac{1}{x}

\frac{d}{d x} (cos (x)) = - sin (x)

= \frac{1}{x} cos (x) + (- sin (x)) ln (x)

Simplify

\frac{1}{x} cos (x) + (- sin (x)) ln (x) : \frac{cos ( x )}{x} - sin (x) ln (x)

= \frac{cos ( x )}{x} - sin (x) ln (x)

What is the derivative of ln(xcos(x)) ?
The derivative of ln(xcos(x)) is (cos(x))/x-sin(x)ln(x)
What is the first derivative of ln(xcos(x)) ?
The first derivative of ln(xcos(x)) is (cos(x))/x-sin(x)ln(x)