Find dy/dx in terms of \( x \& y \) : \( \begin{array}{l}\text { 1. } x y-x+2 y=1 \\ \text { 2. } x^{3}+y^{3}=1 \\ \text { 3. } x^{2}+x y=y^{3} \\ \text { 4. } x^{3} y+x y^{5}=2 \\ \text { 5. } x^{2} y^{3}=2 x-y \\ \text { 6. } x^{2}+4(y-1)^{2}=4 \\ \text { 7. } \frac{x-y}{x+y}=\frac{x^{2}}{y}+1\end{array} \)

Calculate or simplify the expression \( \sin(45) \). Calculate the value by following steps: - step0: Calculate: \(\sin\left(45\right)\) Calculate or simplify the expression \( \cos(\pi/4) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(\frac{\pi }{4}\right)\) - step1: Calculate the trigonometric value: \(\frac{\sqrt{2}}{2}\) Calculate or simplify the expression \( \sin(45)/\cos(\pi/4) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\sin\left(45\right)}{\cos\left(\frac{\pi }{4}\right)}\) - step1: Transform the expression: \(\frac{2\sin\left(45\right)}{\sqrt{2}}\) - step2: Multiply by the Conjugate: \(\frac{2\sin\left(45\right)\times \sqrt{2}}{\sqrt{2}\times \sqrt{2}}\) - step3: Multiply the numbers: \(\frac{2\sqrt{2}\times \sin\left(45\right)}{\sqrt{2}\times \sqrt{2}}\) - step4: Multiply the numbers: \(\frac{2\sqrt{2}\times \sin\left(45\right)}{2}\) - step5: Reduce the fraction: \(\sqrt{2}\times \sin\left(45\right)\) The value of \( \frac{\sin(45^{\circ})}{\cos(\frac{\pi}{4})} \) is \( \sqrt{2} \).