Math Problem Statement

d/dx cubert x

Solution

To differentiate x3\sqrt[3]{x} (which is the cube root of xx) with respect to xx, we can rewrite it as a power of xx.

x3=x1/3\sqrt[3]{x} = x^{1/3}

Now, apply the power rule for differentiation, which states:

ddx(xn)=nxn1\frac{d}{dx} \left( x^n \right) = n \cdot x^{n-1}

In this case, n=13n = \frac{1}{3}. So, we differentiate:

ddx(x1/3)=13x1/31=13x2/3\frac{d}{dx} \left( x^{1/3} \right) = \frac{1}{3} \cdot x^{1/3 - 1} = \frac{1}{3} \cdot x^{-2/3}

Thus, the derivative is:

ddx(x3)=13x2/3\frac{d}{dx} \left( \sqrt[3]{x} \right) = \frac{1}{3} x^{-2/3}

Or equivalently:

131x2/3\frac{1}{3} \cdot \frac{1}{x^{2/3}}

Let me know if you'd like more details or have further questions!


Here are 5 follow-up questions for more exploration:

  1. How do you differentiate other roots, like square roots or higher-order roots of xx?
  2. What would the derivative be if the function was x23\sqrt[3]{x^2}?
  3. How does the power rule work for fractional exponents?
  4. What is the integral of x1/3x^{1/3}?
  5. Can you apply the chain rule to differentiate a composite function like 2x+13\sqrt[3]{2x+1}?

Tip: To handle fractional exponents, remember that xm/n=xmnx^{m/n} = \sqrt[n]{x^m}, which lets you treat roots as powers!

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Math Problem Analysis

Mathematical Concepts

Differentiation
Exponents
Power Rule

Formulas

d/dx(x^n) = n * x^(n-1)
x^(1/n) = nth root of x

Theorems

Power Rule for Differentiation

Suitable Grade Level

Grades 9-11