What is the derivative of sin(xln(5x)) ?

The derivative of sin(xln(5x)) is cos(x)ln(5x)+(sin(x))/x

What is the first derivative of sin(xln(5x)) ?

The first derivative of sin(xln(5x)) is cos(x)ln(5x)+(sin(x))/x

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\frac{d}{d x} (sin (x) ln (5 x))

cos (x) ln (5 x) + \frac{sin ( x )}{x}

Solution steps

\frac{d}{d x} (sin (x) ln (5 x))

Apply the Product Rule:

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

f = sin (x), g = ln (5 x)

= \frac{d}{d x} (sin (x)) ln (5 x) + \frac{d}{d x} (ln (5 x)) sin (x)

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

\frac{d}{d x} (ln (5 x)) = \frac{1}{x}

= cos (x) ln (5 x) + \frac{1}{x} sin (x)

\frac{1}{x} sin (x) = \frac{sin ( x )}{x}

= cos (x) ln (5 x) + \frac{sin ( x )}{x}

What is the derivative of sin(xln(5x)) ?
The derivative of sin(xln(5x)) is cos(x)ln(5x)+(sin(x))/x
What is the first derivative of sin(xln(5x)) ?
The first derivative of sin(xln(5x)) is cos(x)ln(5x)+(sin(x))/x