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

The derivative of xsin(ln(x)) is sin(ln(x))+cos(ln(x))

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

The first derivative of xsin(ln(x)) is sin(ln(x))+cos(ln(x))

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

What is the derivative of xsin(ln(x)) ?
The derivative of xsin(ln(x)) is sin(ln(x))+cos(ln(x))
What is the first derivative of xsin(ln(x)) ?
The first derivative of xsin(ln(x)) is sin(ln(x))+cos(ln(x))