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User:Freezertennis3423 - Wikipedia, the free encyclopedia

User:Freezertennis3423

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Derivative of $ln(x\sqrt{x^2+1})$ :

Using chain rule... Derivative of $x^2+1 \,$ is $2x \,$ , therefore derivative of $\sqrt {x^2+1}$ is $2x {1 \over 2\sqrt {x^2+1}}$ , which simplifies to ${2x \over 2\sqrt {x^2+1}}$ , which simplifies to ${x \over \sqrt {x^2+1}}$

Then using product rule: The derivative of $x\sqrt{x^2+1}$ is $\sqrt{x^2+1}+x{x \over \sqrt {x^2+1}}$ , which simplifies to $\sqrt{x^2+1}+{x^2 \over \sqrt {x^2+1}}$ , which can be rewritten as ${2x^2+1 \over \sqrt {x^2+1}}$ by finding a common denominator

Finally, using the chain rule, the derivative of $ln(x\sqrt{x^2+1}) \,$ is ${2x^2+1 \over \sqrt {x^2+1}} \times {1 \over {x\sqrt{x^2+1}}}$ , which simplifies to ${2x^2+1} \over {x\sqrt{x^2+1}\sqrt{x^2+1}}$ or ${2x^2+1} \over {x(x^2+1)}$ or ${2x^2+1} \over {x^3+x}$ , which can be simplified to ${x \over {x^2+1}}+{1 \over x}$ by using algebra

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