What is the derivative of sin(xln(x^4+1/x)) ?

The derivative of sin(xln(x^4+1/x)) is cos(x)ln(x^4+1/x)+(sin(x)(4x^5-1))/(x(x^5+1))

What is the first derivative of sin(xln(x^4+1/x)) ?

The first derivative of sin(xln(x^4+1/x)) is cos(x)ln(x^4+1/x)+(sin(x)(4x^5-1))/(x(x^5+1))

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\frac{d}{d x} (sin (x) ln (x^{4} + \frac{1}{x}))

cos (x) ln (x^{4} + \frac{1}{x}) + \frac{sin ( x ) ( 4 x ^{5} - 1 )}{x ( x ^{5} + 1 )}

Solution steps

\frac{d}{d x} (sin (x) ln (x^{4} + \frac{1}{x}))

Apply the Product Rule:

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

f = sin (x), g = ln (x^{4} + \frac{1}{x})

= \frac{d}{d x} (sin (x)) ln (x^{4} + \frac{1}{x}) + \frac{d}{d x} (ln (x^{4} + \frac{1}{x})) sin (x)

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

\frac{d}{d x} (ln (x^{4} + \frac{1}{x})) = \frac{4 x ^{5} - 1}{x ( x ^{5} + 1 )}

= cos (x) ln (x^{4} + \frac{1}{x}) + \frac{4 x ^{5} - 1}{x ( x ^{5} + 1 )} sin (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= cos (x) ln (x^{4} + \frac{1}{x}) + \frac{sin ( x ) ( 4 x ^{5} - 1 )}{x ( x ^{5} + 1 )}

f (x) = x^{3} \cdot e^{3 x}

\frac{d}{d x} (\frac{6 x}{x ^{2} + 1})

\frac{d}{d x} (arcsin (sin (11 x)))

\int \frac{1}{t ^{2} - 2 t + 10} d t

\int cos^{7} (2 x) d x

What is the derivative of sin(xln(x^4+1/x)) ?
The derivative of sin(xln(x^4+1/x)) is cos(x)ln(x^4+1/x)+(sin(x)(4x^5-1))/(x(x^5+1))
What is the first derivative of sin(xln(x^4+1/x)) ?
The first derivative of sin(xln(x^4+1/x)) is cos(x)ln(x^4+1/x)+(sin(x)(4x^5-1))/(x(x^5+1))