What is the derivative of ln(sqrt(x^2*e^{x^2+1)}) ?

The derivative of ln(sqrt(x^2*e^{x^2+1)}) is (x|x|+1)/(|x|)

What is the first derivative of ln(sqrt(x^2*e^{x^2+1)}) ?

The first derivative of ln(sqrt(x^2*e^{x^2+1)}) is (x|x|+1)/(|x|)

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

\frac{x ∣ x ∣ + 1}{∣ x ∣}

Solution steps

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

Simplify

ln (x^{2} e^{x^{2} + 1}) : ln (x^{2} e^{x^{2} + 1})

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

Apply the chain rule:

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

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

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

= \frac{1}{x ^{2} e ^{x^{2} + 1}} (\frac{x e ^{x^{2} + 1}}{∣ x ∣} + \frac{e ^{x^{2} + 1} x x ^{2}}{e ^{x^{2} + 1}})

Simplify

\frac{1}{x ^{2} e ^{x^{2} + 1}} (\frac{x e ^{x^{2} + 1}}{∣ x ∣} + \frac{e ^{x^{2} + 1} x x ^{2}}{e ^{x^{2} + 1}}) : \frac{x ∣ x ∣ + 1}{∣ x ∣}

= \frac{x ∣ x ∣ + 1}{∣ x ∣}

What is the derivative of ln(sqrt(x^2*e^{x^2+1)}) ?
The derivative of ln(sqrt(x^2*e^{x^2+1)}) is (x|x|+1)/(|x|)
What is the first derivative of ln(sqrt(x^2*e^{x^2+1)}) ?
The first derivative of ln(sqrt(x^2*e^{x^2+1)}) is (x|x|+1)/(|x|)