Find the derivatives of the functions in Exercises 1–42.
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𝔂 = / x² + x
√ x²

Verified step by step guidance

1. First, simplify the given function \( y = \frac{1}{x^2} + x\sqrt{x^2} \). Notice that \( \sqrt{x^2} = |x| \), but for simplicity, we can assume \( x \geq 0 \) so \( \sqrt{x^2} = x \). Thus, the function becomes \( y = \frac{1}{x^2} + x^2 \).
2. To find the derivative \( y' \), apply the sum rule of differentiation, which states that the derivative of a sum is the sum of the derivatives. Therefore, differentiate each term separately.
3. For the first term \( \frac{1}{x^2} \), rewrite it as \( x^{-2} \) and use the power rule for differentiation: \( \frac{d}{dx} x^n = nx^{n-1} \). Thus, the derivative is \( -2x^{-3} \).
4. For the second term \( x^2 \), again use the power rule: \( \frac{d}{dx} x^2 = 2x \).
5. Combine the derivatives of each term to find the derivative of the entire function: \( y' = -2x^{-3} + 2x \).