Exercise
Find the derivative of the following function:
f ( x ) = x 4 − x 2 + 2 x x f(x)=\frac{x^4-x^2+2}{x\sqrt{x}} f ( x ) = x x x 4 − x 2 + 2
Final Answer
Show final answer
f'(x)=\frac{5x^4-x^2+6}{2x^2\sqrt{x}}
Solution
We simplify the function before differentiating:
f ( x ) = x 4 − x 2 + 2 x x = f(x)=\frac{x^4-x^2+2}{x\sqrt{x}}= f ( x ) = x x x 4 − x 2 + 2 =
f ( x ) = x 4 − x 2 + 2 x 3 2 f(x)=\frac{x^4-x^2+2}{x^{\frac{3}{2}}} f ( x ) = x 2 3 x 4 − x 2 + 2
Using Derivative formulas Derivative Rules
f ′ ( x ) = ( 4 x 3 − 2 x ) ⋅ x x − ( x 4 − x 2 + 2 ) ⋅ 3 2 x ( x x ) 2 = f'(x)=\frac{(4x^3-2x)\cdot x\sqrt{x}-(x^4-x^2+2)\cdot \frac{3}{2}\sqrt{x}}{{(x\sqrt{x})}^2}= f ′ ( x ) = ( x x ) 2 ( 4 x 3 − 2 x ) ⋅ x x − ( x 4 − x 2 + 2 ) ⋅ 2 3 x =
One can simplify the derivative:
= 4 x 4 x − 2 x 2 x − 3 2 x 4 x + 3 2 x 2 x + 3 x x 2 ⋅ x = =\frac{4x^4\sqrt{x}-2x^2\sqrt{x}-\frac{3}{2}x^4\sqrt{x}+\frac{3}{2}x^2\sqrt{x}+3\sqrt{x}}{x^2\cdot x}= = x 2 ⋅ x 4 x 4 x − 2 x 2 x − 2 3 x 4 x + 2 3 x 2 x + 3 x =
= 5 2 x 4 x − 1 2 x 2 x + 3 x x 3 = =\frac{\frac{5}{2}x^4\sqrt{x}-\frac{1}{2}x^2\sqrt{x}+3\sqrt{x}}{x^3}= = x 3 2 5 x 4 x − 2 1 x 2 x + 3 x =
= 5 x 4 x − x 2 x + 6 x 2 x 3 = =\frac{5x^4\sqrt{x}-x^2\sqrt{x}+6\sqrt{x}}{2x^3}= = 2 x 3 5 x 4 x − x 2 x + 6 x =
= 5 x 4 − x 2 + 6 2 x 2 x =\frac{5x^4-x^2+6}{2x^2\sqrt{x}} = 2 x 2 x 5 x 4 − x 2 + 6
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