What is the derivative of ln(sqrt(x^2+5)) ?

The derivative of ln(sqrt(x^2+5)) is x/(x^2+5)

What is the first derivative of ln(sqrt(x^2+5)) ?

The first derivative of ln(sqrt(x^2+5)) is x/(x^2+5)

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\frac{d}{d x} (ln (x^{2} + 5))

\frac{x}{x ^{2} + 5}

Solution steps

\frac{d}{d x} (ln (x^{2} + 5))

Apply the chain rule:

\frac{1}{x ^{2} + 5} \frac{d}{d x} (x^{2} + 5)

= \frac{1}{x ^{2} + 5} \frac{d}{d x} (x^{2} + 5)

\frac{d}{d x} (x^{2} + 5) = \frac{x}{x ^{2} + 5}

= \frac{1}{x ^{2} + 5} \cdot \frac{x}{x ^{2} + 5}

Simplify

\frac{1}{x ^{2} + 5} \cdot \frac{x}{x ^{2} + 5} : \frac{x}{x ^{2} + 5}

= \frac{x}{x ^{2} + 5}

What is the derivative of ln(sqrt(x^2+5)) ?
The derivative of ln(sqrt(x^2+5)) is x/(x^2+5)
What is the first derivative of ln(sqrt(x^2+5)) ?
The first derivative of ln(sqrt(x^2+5)) is x/(x^2+5)