What is the derivative of ln(sqrt((3x^4+2)/(4x))) ?

The derivative of ln(sqrt((3x^4+2)/(4x))) is (9x^4-2)/(2x(3x^4+2))

What is the first derivative of ln(sqrt((3x^4+2)/(4x))) ?

The first derivative of ln(sqrt((3x^4+2)/(4x))) is (9x^4-2)/(2x(3x^4+2))

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derivative of

ln (\frac{3 x ^{4} + 2}{4 x})

\frac{9 x ^{4} - 2}{2 x ( 3 x ^{4} + 2 )}

Solution steps

\frac{d}{d x} (ln (\frac{3 x ^{4} + 2}{4 x}))

Simplify

ln (\frac{3 x ^{4} + 2}{4 x}) : ln (\frac{3 x ^{4} + 2}{2 x})

= \frac{d}{d x} (ln (\frac{3 x ^{4} + 2}{2 x}))

Apply the chain rule:

\frac{1}{\frac{3 x ^{4} + 2}{2 x}} \frac{d}{d x} (\frac{3 x ^{4} + 2}{2 x})

= \frac{1}{\frac{3 x ^{4} + 2}{2 x}} \frac{d}{d x} (\frac{3 x ^{4} + 2}{2 x})

\frac{d}{d x} (\frac{3 x ^{4} + 2}{2 x}) = \frac{9 x ^{4} - 2}{4 x x 3 x ^{4} + 2}

= \frac{1}{\frac{3 x ^{4} + 2}{2 x}} \cdot \frac{9 x ^{4} - 2}{4 x x 3 x ^{4} + 2}

Simplify

\frac{1}{\frac{3 x ^{4} + 2}{2 x}} \cdot \frac{9 x ^{4} - 2}{4 x x 3 x ^{4} + 2} : \frac{9 x ^{4} - 2}{2 x ( 3 x ^{4} + 2 )}

= \frac{9 x ^{4} - 2}{2 x ( 3 x ^{4} + 2 )}

\frac{\partial}{\partial x} (y^{3} e^{2 x} + 5 sin (x y))

\frac{d}{d x} (x^{2} + 3 x - x)

y^{^{''}} + 6 y = 0, y^{^{'}} (1) = 5

\frac{d y}{d x} = \frac{( x ^{2} + 1 )}{y ^{2} + 1}, y (0) = 1

\frac{d}{d x} (arccos (1 - 2 x^{2}))

What is the derivative of ln(sqrt((3x^4+2)/(4x))) ?
The derivative of ln(sqrt((3x^4+2)/(4x))) is (9x^4-2)/(2x(3x^4+2))
What is the first derivative of ln(sqrt((3x^4+2)/(4x))) ?
The first derivative of ln(sqrt((3x^4+2)/(4x))) is (9x^4-2)/(2x(3x^4+2))