What is the derivative of ln(sqrt(x^2+7)) ?

The derivative of ln(sqrt(x^2+7)) is x/(x^2+7)

What is the first derivative of ln(sqrt(x^2+7)) ?

The first derivative of ln(sqrt(x^2+7)) is x/(x^2+7)

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\frac{d}{d x} (ln (x^{2} + 7))

\frac{x}{x ^{2} + 7}

Solution steps

\frac{d}{d x} (ln (x^{2} + 7))

Apply the chain rule:

\frac{1}{x ^{2} + 7} \frac{d}{d x} (x^{2} + 7)

= \frac{1}{x ^{2} + 7} \frac{d}{d x} (x^{2} + 7)

\frac{d}{d x} (x^{2} + 7) = \frac{x}{x ^{2} + 7}

= \frac{1}{x ^{2} + 7} \cdot \frac{x}{x ^{2} + 7}

Simplify

\frac{1}{x ^{2} + 7} \cdot \frac{x}{x ^{2} + 7} : \frac{x}{x ^{2} + 7}

= \frac{x}{x ^{2} + 7}

What is the derivative of ln(sqrt(x^2+7)) ?
The derivative of ln(sqrt(x^2+7)) is x/(x^2+7)
What is the first derivative of ln(sqrt(x^2+7)) ?
The first derivative of ln(sqrt(x^2+7)) is x/(x^2+7)