Determine
[tex]D_x\left[\frac{\sqrt[3]{x^2}-x^{-\frac{3}{2}}}{\sqrt{x}}\right][/tex]
Answer (1)
Rewrite the function using exponent rules: f ( x ) = x 6 1 − x − 2 .
Differentiate each term using the power rule: D x [ x n ] = n x n − 1 .
Obtain the derivative: f ′ ( x ) = 6 1 x − 6 5 + 2 x − 3 .
Simplify the expression: x 3.0 2.0 + x 0.833333333333334 0.166666666666667 .
Explanation
Problem Setup We are asked to find the derivative of the function f ( x ) = x 3 x 2 − x − 2 3 with respect to x . This is denoted as D x [ f ( x )] .
Rewriting the Function First, let's rewrite the function using exponent rules to make it easier to differentiate: f ( x ) = x 2 1 x 3 2 − x − 2 3 = x 3 2 − 2 1 − x − 2 3 − 2 1 = x 6 4 − 6 3 − x − 2 4 = x 6 1 − x − 2
Differentiating Each Term Now, we can differentiate each term using the power rule, which states that if f ( x ) = x n , then f ′ ( x ) = n x n − 1 . Applying this rule, we get: D x [ x 6 1 ] = 6 1 x 6 1 − 1 = 6 1 x − 6 5 D x [ x − 2 ] = − 2 x − 2 − 1 = − 2 x − 3
Combining the Derivatives Therefore, the derivative of the function is: f ′ ( x ) = 6 1 x − 6 5 − ( − 2 x − 3 ) = 6 1 x − 6 5 + 2 x − 3 We can rewrite this as: f ′ ( x ) = 6 x 6 5 1 + x 3 2
Simplified Form To express the answer in a simplified form, we can write: f ′ ( x ) = 6 x 6 5 1 + x 3 2 = x 3.0 2.0 + x 0.833333333333334 0.166666666666667
Final Answer Thus, the derivative of the given function is: x 3.0 2.0 + x 0.833333333333334 0.166666666666667
Examples
In physics, if x ( t ) represents the displacement of a particle at time t , then the derivative x ′ ( t ) gives the velocity of the particle. If the displacement is given by a function similar to the one in this problem, finding its derivative would allow us to determine the particle's velocity at any given time. This is crucial in understanding the motion of objects under various forces.
