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

The derivative of xln(xcos(x)) is ln(x)cos(x)+cos(x)-xsin(x)ln(x)

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

The first derivative of xln(xcos(x)) is ln(x)cos(x)+cos(x)-xsin(x)ln(x)

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

ln (x) cos (x) + cos (x) - x sin (x) ln (x)

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

= ln (x) cos (x) + cos (x) - x sin (x) ln (x)

What is the derivative of xln(xcos(x)) ?
The derivative of xln(xcos(x)) is ln(x)cos(x)+cos(x)-xsin(x)ln(x)
What is the first derivative of xln(xcos(x)) ?
The first derivative of xln(xcos(x)) is ln(x)cos(x)+cos(x)-xsin(x)ln(x)