Math Problem Statement

what is the derivative of f(x)=2/x

Solution

The function you provided is:

f(x)=2xf(x) = \frac{2}{x}

To find the derivative f(x)f'(x), we can rewrite the function as:

f(x)=2x1f(x) = 2x^{-1}

Now, apply the power rule of differentiation, which states that the derivative of xnx^n is nxn1n \cdot x^{n-1}. For 2x12x^{-1}, the derivative is:

f(x)=2(1)x2=2x2f'(x) = 2 \cdot (-1) \cdot x^{-2} = -\frac{2}{x^2}

Thus, the derivative of f(x)=2xf(x) = \frac{2}{x} is:

f(x)=2x2f'(x) = -\frac{2}{x^2}

Would you like more details or have any questions?

Here are 5 related questions to explore further:

  1. What is the derivative of f(x)=5xf(x) = \frac{5}{x}?
  2. How do you find the derivative of a quotient of two functions?
  3. Can you explain the product rule for derivatives?
  4. What is the second derivative of f(x)=2xf(x) = \frac{2}{x}?
  5. How would you integrate f(x)=2xf(x) = \frac{2}{x}?

Tip: Rewriting functions in terms of exponents often makes differentiation easier.

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Power Rule

Formulas

f(x) = 2x^{-1}
Power Rule: d/dx[x^n] = n * x^{n-1}

Theorems

Power Rule of Differentiation

Suitable Grade Level

Grades 11-12