Exercise
Find the derivative of the following function:
f ( x ) = x 2 x 2 − 9 − 9 2 ln ( x + x 2 − 9 ) f(x)=\frac{x}{2}\sqrt{x^2-9}-\frac{9}{2}\ln(x+\sqrt{x^2-9}) f ( x ) = 2 x x 2 − 9 − 2 9 ln ( x + x 2 − 9 )
Final Answer
Show final answer
f'(x)=\sqrt{x^2-9}
Solution
f ( x ) = x 2 x 2 − 9 − 9 2 ln ( x + x 2 − 9 ) f(x)=\frac{x}{2}\sqrt{x^2-9}-\frac{9}{2}\ln(x+\sqrt{x^2-9}) f ( x ) = 2 x x 2 − 9 − 2 9 ln ( x + x 2 − 9 )
Using Derivative formulas Derivative Rules
f ′ ( x ) = 1 2 x 2 − 9 + x 2 ⋅ 2 x 2 x 2 − 9 − 9 2 ⋅ 1 x + x 2 − 9 ⋅ ( 1 + 1 2 x 2 − 9 ⋅ 2 x ) = f'(x)=\frac{1}{2}\sqrt{x^2-9}+\frac{x}{2}\cdot\frac{2x}{2\sqrt{x^2-9}}-\frac{9}{2}\cdot\frac{1}{x+\sqrt{x^2-9}}\cdot (1+\frac{1}{2\sqrt{x^2-9}}\cdot 2x)= f ′ ( x ) = 2 1 x 2 − 9 + 2 x ⋅ 2 x 2 − 9 2 x − 2 9 ⋅ x + x 2 − 9 1 ⋅ ( 1 + 2 x 2 − 9 1 ⋅ 2 x ) =
One can simplify the derivative:
= x 2 − 9 2 + x 2 2 x 2 − 9 − 9 2 ⋅ 1 x + x 2 − 9 ⋅ x 2 − 9 + x x 2 − 9 = =\frac{\sqrt{x^2-9}}{2}+\frac{x^2}{2\sqrt{x^2-9}}-\frac{9}{2}\cdot\frac{1}{x+\sqrt{x^2-9}}\cdot\frac{\sqrt{x^2-9}+x}{\sqrt{x^2-9}}= = 2 x 2 − 9 + 2 x 2 − 9 x 2 − 2 9 ⋅ x + x 2 − 9 1 ⋅ x 2 − 9 x 2 − 9 + x =
= x 2 − 9 2 + x 2 2 x 2 − 9 − 9 2 x 2 − 9 = =\frac{\sqrt{x^2-9}}{2}+\frac{x^2}{2\sqrt{x^2-9}}-\frac{9}{2\sqrt{x^2-9}}= = 2 x 2 − 9 + 2 x 2 − 9 x 2 − 2 x 2 − 9 9 =
= x 2 − 9 + x 2 − 9 2 x 2 − 9 = =\frac{x^2-9+x^2-9}{2\sqrt{x^2-9}}= = 2 x 2 − 9 x 2 − 9 + x 2 − 9 =
= 2 ( x 2 − 9 ) 2 x 2 − 9 = =\frac{2(x^2-9)}{2\sqrt{x^2-9}}= = 2 x 2 − 9 2 ( x 2 − 9 ) =
= x 2 − 9 =\sqrt{x^2-9} = x 2 − 9
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