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Calculating Derivative – A multiplication of a polynom and square root – Exercise 6352

Exercise

Find the derivative of the following function:

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

Final Answer


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

Solution

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

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

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

One can simplify the derivative:

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

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

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

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

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