Math Problem Statement

Find the derivative of: 7√x − 3/x³ − 3x + 6

Solution

The given expression is:

f(x)=7x33x33x+6f(x) = 7 \sqrt[3]{x} - \frac{3}{x^3} - 3x + 6

We need to find the derivative f(x)f'(x).

Step-by-step solution:

  1. Rewrite the terms in exponent form:
    • 7x37 \sqrt[3]{x} becomes 7x1/37x^{1/3}
    • 3x3\frac{3}{x^3} becomes 3x33x^{-3}
    • The term 3x-3x remains as is.
    • The constant 66 remains unchanged (its derivative will be 0).

So the function now looks like: f(x)=7x1/33x33x+6f(x) = 7x^{1/3} - 3x^{-3} - 3x + 6

  1. Differentiate each term:

    • For 7x1/37x^{1/3}, apply the power rule: ddx(xn)=nxn1\frac{d}{dx}(x^n) = n x^{n-1} ddx(7x1/3)=713x1/31=73x2/3\frac{d}{dx}(7x^{1/3}) = 7 \cdot \frac{1}{3} x^{1/3 - 1} = \frac{7}{3}x^{-2/3}
    • For 3x3-3x^{-3}, apply the power rule again: ddx(3x3)=3(3)x4=9x4\frac{d}{dx}(-3x^{-3}) = -3 \cdot (-3) x^{-4} = 9x^{-4}
    • For 3x-3x, the derivative is simply: ddx(3x)=3\frac{d}{dx}(-3x) = -3
    • The derivative of the constant 66 is 0.
  2. Combine the results: f(x)=73x2/3+9x43f'(x) = \frac{7}{3}x^{-2/3} + 9x^{-4} - 3

Thus, the derivative of the given function is:

f(x)=73x2/3+9x43f'(x) = \frac{7}{3}x^{-2/3} + 9x^{-4} - 3

Would you like more details on any specific part of this solution?

5 Related Questions:

  1. How do you apply the power rule to expressions with fractional exponents?
  2. What is the derivative of a cube root function like x3\sqrt[3]{x}?
  3. How can you differentiate rational expressions like 1x3\frac{1}{x^3}?
  4. Why is the derivative of a constant always zero?
  5. Can you explain the general rules for differentiating terms with negative exponents?

Tip:

When differentiating functions with roots or fractional exponents, always rewrite them in exponent form to simplify the application of the power rule.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Power Rule
Fractional Exponents
Negative Exponents

Formulas

Power rule: d/dx(x^n) = n*x^(n-1)
Fractional exponent rule: d/dx(x^(m/n)) = (m/n)*x^((m/n)-1)
Negative exponent rule: d/dx(x^(-n)) = -n*x^(-n-1)

Theorems

Power Rule

Suitable Grade Level

Grades 11-12 (Calculus)