What is the derivative of xln(cos(x)) ?

The derivative of xln(cos(x)) is ln(cos(x))-xtan(x)

What is the first derivative of xln(cos(x)) ?

The first derivative of xln(cos(x)) is ln(cos(x))-xtan(x)

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

ln (cos (x)) - x tan (x)

Solution steps

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

Apply the Product Rule:

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

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

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

\frac{d x}{d x} = 1

\frac{d}{d x} (ln (cos (x))) = - tan (x)

= 1 \cdot ln (cos (x)) + (- tan (x)) x

Simplify

= ln (cos (x)) - x tan (x)

What is the derivative of xln(cos(x)) ?
The derivative of xln(cos(x)) is ln(cos(x))-xtan(x)
What is the first derivative of xln(cos(x)) ?
The first derivative of xln(cos(x)) is ln(cos(x))-xtan(x)