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1. \frac{d}{dx}\left[\log{\left(x^{2} + y^{2} \right)}\right]

   Differentiate 2 times, one derivative at a time.

2. \frac{\frac{\partial}{\partial x} \left(x^{2} + y^{2}\right)}{x^{2} + y^{2}}

   Chain rule: (ln u)′ = 1/u, times u′ where u = x^{2} + y^{2}.

3. \frac{\frac{d}{d x} x^{2} + \frac{d}{d x} y^{2}}{x^{2} + y^{2}}

   Sum rule: differentiate term by term.

4. \frac{2 x + \frac{d}{d x} y^{2}}{x^{2} + y^{2}}

   Power rule: (xⁿ)′ = n·xⁿ⁻¹ with n = 2.

5. \frac{2 x}{x^{2} + y^{2}}

   The derivative of a constant is 0.

6. f^{(1)}(x) = \frac{2 x}{x^{2} + y^{2}}

   That is derivative number 1.

7. 2 \frac{\partial}{\partial x} \frac{x}{x^{2} + y^{2}}

   Constant multiple rule: pull the constant out.

8. 2 x \frac{\partial}{\partial x} \frac{1}{x^{2} + y^{2}} + \frac{2 \frac{d}{d x} x}{x^{2} + y^{2}}

   Product rule with u = x and v = \frac{1}{x^{2} + y^{2}}: (uv)′ = u′v + uv′.

9. - \frac{2 x \frac{\partial}{\partial x} \left(x^{2} + y^{2}\right)}{\left(x^{2} + y^{2}\right)^{2}} + \frac{2 \frac{d}{d x} x}{x^{2} + y^{2}}

   Reciprocal rule: (1/v)′ = −v′/v² with v = x^{2} + y^{2}.

10. - \frac{2 x \frac{\partial}{\partial x} \left(x^{2} + y^{2}\right)}{\left(x^{2} + y^{2}\right)^{2}} + \frac{2}{x^{2} + y^{2}}

    d/dx of x is 1.

11. - \frac{2 x \left(\frac{d}{d x} x^{2} + \frac{d}{d x} y^{2}\right)}{\left(x^{2} + y^{2}\right)^{2}} + \frac{2}{x^{2} + y^{2}}

    Sum rule: differentiate term by term.

12. - \frac{2 x \left(2 x + \frac{d}{d x} y^{2}\right)}{\left(x^{2} + y^{2}\right)^{2}} + \frac{2}{x^{2} + y^{2}}

    Power rule: (xⁿ)′ = n·xⁿ⁻¹ with n = 2.

13. - \frac{4 x^{2}}{\left(x^{2} + y^{2}\right)^{2}} + \frac{2}{x^{2} + y^{2}}

    The derivative of a constant is 0.

14. f^{(2)}(x) = - \frac{4 x^{2}}{\left(x^{2} + y^{2}\right)^{2}} + \frac{2}{x^{2} + y^{2}}

    That is derivative number 2.