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

The derivative of ln(sqrt(h(x^2))) is 1/(|x|)

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

The first derivative of ln(sqrt(h(x^2))) is 1/(|x|)

\frac{d}{d x} (ln (h (x^{2})))

\frac{1}{∣ x ∣}

Solution steps

\frac{d}{d x} (ln (h (x^{2})))

Simplify

ln (h (x^{2})) : ln (h x^{2})

= \frac{d}{d x} (ln (h x^{2}))

Treat

h

as a constant

Apply the chain rule:

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

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

\frac{d}{d x} (h x^{2}) = \frac{h x}{∣ x ∣}

= \frac{1}{h x ^{2}} \cdot \frac{h x}{∣ x ∣}

Simplify

\frac{1}{h x ^{2}} \cdot \frac{h x}{∣ x ∣} : \frac{1}{∣ x ∣}

= \frac{1}{∣ x ∣}

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