Exercise
Find the derivative of the following function:
f ( x ) = ( x − 1 ) x 2 + 1 f(x)=(x-1)\sqrt{x^2+1} f ( x ) = ( x − 1 ) x 2 + 1
Final Answer
Show final answer
f'(x)=\frac{2x^2-x+1}{\sqrt{x^2+1}}
Solution
f ( x ) = ( x − 1 ) x 2 + 1 f(x)=(x-1)\sqrt{x^2+1} f ( x ) = ( x − 1 ) x 2 + 1
Using Derivative formulas Derivative Rules
f ′ ( x ) = 1 ⋅ x 2 + 1 + ( x − 1 ) ⋅ 1 2 x 2 + 1 ⋅ 2 x = f'(x)=1\cdot\sqrt{x^2+1}+(x-1)\cdot\frac{1}{2\sqrt{x^2+1}}\cdot 2x= f ′ ( x ) = 1 ⋅ x 2 + 1 + ( x − 1 ) ⋅ 2 x 2 + 1 1 ⋅ 2 x =
One can simplify the derivative:
= x 2 + 1 + x ( x − 1 ) x 2 + 1 = =\sqrt{x^2+1}+\frac{x(x-1)}{\sqrt{x^2+1}}= = x 2 + 1 + x 2 + 1 x ( x − 1 ) =
= x 2 + 1 + x ( x − 1 ) x 2 + 1 = =\frac{x^2+1+x(x-1)}{\sqrt{x^2+1}}= = x 2 + 1 x 2 + 1 + x ( x − 1 ) =
= x 2 + 1 + x 2 − x x 2 + 1 = =\frac{x^2+1+x^2-x}{\sqrt{x^2+1}}= = x 2 + 1 x 2 + 1 + x 2 − x =
= 2 x 2 − x + 1 x 2 + 1 =\frac{2x^2-x+1}{\sqrt{x^2+1}} = x 2 + 1 2 x 2 − x + 1
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