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Logarithmic differentiation is a powerful technique to differentiate functions of the form $\text{f}(\text{x})=[\text{u}(\text{x})]^{\text{v}(\text{x})},$ where both u(x) and v(x) are differentiable functions and f and u need to be positive functions. Let function $\text{y}=\text{f}(\text{x})=(\text{u}(\text{x}))^{\text{v}(\text{x})},$ then $\text{y}'=\text{y}\Big[\frac{\text{v}(\text{x})}{\text{u}(\text{x})}\text{u}'(\text{x})+\text{v}'(\text{x})\cdot\log[\text{u}(\text{x})]\Big]$ On the basis of above information, answer the following questions.

1. Differentiate $x^x$ w.r.t. x.

1. $\text{x}^\text{x}(1+\log\text{x})$
2. $\text{x}^\text{x}(1-\log\text{x})$
3. $-\text{x}^\text{x}(1+\log\text{x})$
4. $\text{x}^\text{x}\log\text{x}$

2. Differentiate $x^x + a^x+ x^a + a^a$ w.r.t. x.

1. $(1+\log\text{x})+(\text{a}^\text{x}\log\text{a}+\text{ax}^{\text{a}-1})$
2. $\text{x}^\text{x}(1+\log\text{x})+\log\text{a}+\text{ax}^{\text{a}-1}$
3. $\text{x}^\text{x}(1+\log\text{x})+\text{x}^\text{a}\log\text{x}+\text{ax}^{\text{a}-1}$
4. $\text{x}^\text{x}(1+\log\text{x})+\text{a}^\text{x}\log\text{a}+\text{ax}^{\text{a}-1}$

3. If $\text{x}=\text{e}^\frac{\text{x}}{\text{y}},$ then find $\frac{\text{dy}}{\text{dx}}.$

1. $-\frac{(\text{x}+\text{y})}{\text{x}\log\text{x}}$
2. $-\frac{(\text{x}-\text{y})}{\text{x}\log\text{x}}$
3. $\frac{(\text{x}+\text{y})}{\text{x}\log\text{x}}$
4. $\frac{\text{x}-\text{y}}{\text{x}\log\text{x}}$

4. If $y = (2 - x)^3(3 + 2x)^5$, then find $\frac{\text{dy}}{\text{dx}}.$

1. $(2-\text{x})^3(3+2\text{x})^5\Big[\frac{15}{3+2\text{x}}-\frac{8}{2-\text{x}}\Big]$
2. $(2-\text{x})^3(3+2\text{x})^5\Big[\frac{15}{3+2\text{x}}+\frac{3}{2-\text{x}}\Big]$
3. $(2-\text{x})^3(3+2\text{x})^5\Big[\frac{10}{3+2\text{x}}-\frac{3}{2-\text{x}}\Big]$
4. $(2-\text{x})^3(3+2\text{x})^5\cdot\Big[\frac{10}{3+2\text{x}}+\frac{3}{2-\text{x}}\Big]$

5. If $\text{y}=\text{x}^\text{x}\cdot\text{e}^{(2\text{x}+5)},$ then find $\frac{\text{dy}}{\text{dx}}.$

1. $\text{x}^\text{x}\text{e}^{2\text{x}+5}$
2. $\text{x}^\text{x}\text{e}^{2\text{x}+5}(3-\log\text{x})$
3. $\text{x}^\text{x}\text{e}^{2\text{x}+5}(1-\log\text{x})$
4. $\text{x}^\text{x}\text{e}^{2\text{x}+5}\cdot(3+\log\text{x})$