Determine [tex]D_x\left[\frac{\sqrt[3]{x^2}-x^{-\frac{3}{2}}}{\sqrt{x}}\right][/tex]
Answer (2)
Rewrite the function using exponents: f ( x ) = x 6 1 − x − 2 .
Apply the power rule to find the derivative: f ′ ( x ) = 6 1 x − 6 5 + 2 x − 3 .
Rewrite the derivative using radicals and fractions: f ′ ( x ) = 6 6 x 5 1 + x 3 2 .
The derivative of the function is: 6 1 x − 6 5 + 2 x − 3 .
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 can be written as D_x\left[\frac{\sqrt[3]{x^2}-x^{-\frac{3}{2}}}{\sqrt{x}}\] .
Rewriting the Function First, let's rewrite the function using exponents: f ( x ) = x 2 1 x 3 2 − x − 2 3 Now, divide each term in the numerator by x 2 1 :
f ( x ) = x 3 2 − 2 1 − x − 2 3 − 2 1 f ( x ) = x 6 4 − 6 3 − x − 2 4 f ( x ) = x 6 1 − x − 2
Finding the Derivative Now, we can find the derivative of f ( x ) with respect to x using the power rule: f ′ ( x ) = d x d ( x 6 1 − x − 2 ) f ′ ( x ) = 6 1 x 6 1 − 1 − ( − 2 ) x − 2 − 1 f ′ ( x ) = 6 1 x − 6 5 + 2 x − 3
Rewriting the Derivative Rewrite the derivative using radicals and fractions: f ′ ( x ) = 6 x 6 5 1 + x 3 2 f ′ ( x ) = 6 6 x 5 1 + x 3 2
Simplifying (Optional) Combining the terms into a single fraction: f ′ ( x ) = 6 x 6 5 1 + x 3 2 = 6 x 6 5 x 3 x 3 + 12 x 6 5 = 6 x 6 23 x 3 + 12 x 6 5 However, let's stick to the uncombined form for simplicity.
Final Answer The derivative of the given function is: f ′ ( x ) = 6 1 x − 6 5 + 2 x − 3 = 6 6 x 5 1 + x 3 2
Examples
In physics, when analyzing the motion of an object, you might encounter functions involving fractional exponents, especially when dealing with resistive forces or potential energies. Taking the derivative of such a function, as we did here, helps determine the object's velocity or acceleration at a given time. For example, if the position of a particle is given by f ( t ) = t 6 1 − t − 2 , then its velocity is given by the derivative f ′ ( t ) = 6 1 t − 6 5 + 2 t − 3 . This allows physicists to understand and predict the behavior of the particle.
The derivative of the function f ( x ) = x 3 x 2 − x − 2 3 is f ′ ( x ) = 6 1 x − 6 5 + 2 x − 3 = 6 6 x 5 1 + x 3 2 . This is found by rewriting the function using exponents and applying the power rule for differentiation. The final result expresses the derivative in both exponent and fraction forms.
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