Question5. (a) If $y=3 \sin x+x^{2} \ln x$, find $\frac{d y}{d x}$ (b) If $\frac{x^{2}+3 y^{2}}{2 x}=x^{2}$, find $\frac{d y}{d x}$ (c) If $x=\cos \theta+\theta$ and $y=\cos \theta-\theta \quad$, find $d y / d x$ (d) Solve the equation $5^{x+1}=2^{x-1}$

Studdy Solution
Simplify the result: $$x = \frac{- (\ln 2 + \ln 5)}{\ln 5 - \ln 2}$$ $$x = \frac{- \ln (10)}{\ln \left(\frac{5}{2}\right)}$$
Solution: (a) $\frac{d y}{d x} = 3 \cos x + 2x \ln x + x$ (b) $\frac{d y}{d x} = \frac{x (3 x - 1)}{3 y}$ (c) $\frac{d y}{d x} = \frac{\sin \theta + 1}{\sin \theta - 1}$ (d) $x = \frac{- \ln (10)}{\ln \left(\frac{5}{2}\right)}$