What is the derivative of ln(x*sqrt(x^2-1)) ?

The derivative of ln(x*sqrt(x^2-1)) is (2x^2-1)/(x(x^2-1))

What is the first derivative of ln(x*sqrt(x^2-1)) ?

The first derivative of ln(x*sqrt(x^2-1)) is (2x^2-1)/(x(x^2-1))

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

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

Solution steps

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

Apply the chain rule:

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

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

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

= \frac{1}{x x ^{2} - 1} \cdot \frac{2 x ^{2} - 1}{x ^{2} - 1}

Simplify

\frac{1}{x x ^{2} - 1} \cdot \frac{2 x ^{2} - 1}{x ^{2} - 1} : \frac{2 x ^{2} - 1}{x ( x ^{2} - 1 )}

= \frac{2 x ^{2} - 1}{x ( x ^{2} - 1 )}

f (x) = x^{9} cos (x)

t^{2} e^{- 3 t}

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

(0, - 2), (2, 2)

\int_{0}^{1} x + 16 d x

What is the derivative of ln(x*sqrt(x^2-1)) ?
The derivative of ln(x*sqrt(x^2-1)) is (2x^2-1)/(x(x^2-1))
What is the first derivative of ln(x*sqrt(x^2-1)) ?
The first derivative of ln(x*sqrt(x^2-1)) is (2x^2-1)/(x(x^2-1))