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Calculating Derivative – Square root in a ln function – Exercise 6365

Exercise

Find the derivative of the following function:

f(x)=\ln (x+\sqrt{x^2+1})

Final Answer


f'(x)=\frac{1}{\sqrt{x^2+1}}

Solution

f(x)=\ln (x+\sqrt{x^2+1})

Using Derivative formulas and the chain rule and chain rule in Derivative Rules, we get the derivative:

f'(x)=\frac{1}{x+\sqrt{x^2+1}}\cdot (1+\frac{1}{2\sqrt{x^2+1}}\cdot 2x)=

One can simplify the derivative:

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

=\frac{1}{\sqrt{x^2+1}}

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