1. Problem 22: Find $\frac{dy}{dx}$ if $y = (\sin x)^x + \sin^{-1} \sqrt{x}$.\n\n2. To differentiate $y = (\sin x)^x$, use logarithmic differentiation:\n\n$$y = (\sin x)^x \implies \ln y = x \ln(\sin x)$$\n\nDifferentiate both sides w.r.t. $x$:\n\n$$\frac{1}{y} \frac{dy}{dx} = \ln(\sin x) + x \frac{1}{\sin x} \cos x = \ln(\sin x) + x \cot x$$\n\nSo,\n\n$$\frac{dy}{dx} = y \left( \ln(\sin x) + x \cot x \right) = (\sin x)^x \left( \ln(\sin x) + x \cot x \right)$$\n\n3. Differentiate $\sin^{-1} \sqrt{x}$ using chain rule:\n\n$$\frac{d}{dx} \sin^{-1} \sqrt{x} = \frac{1}{\sqrt{1 - (\sqrt{x})^2}} \cdot \frac{d}{dx} \sqrt{x} = \frac{1}{\sqrt{1 - x}} \cdot \frac{1}{2 \sqrt{x}} = \frac{1}{2 \sqrt{x} \sqrt{1 - x}}$$\n\n4. Combine both derivatives:\n\n$$\frac{dy}{dx} = (\sin x)^x \left( \ln(\sin x) + x \cot x \right) + \frac{1}{2 \sqrt{x} \sqrt{1 - x}}$$\n\n---\n\n5. Problem 23: Prove $f(x) = \lfloor x \rfloor$ is not differentiable at $x=2$ for $0 < x < 3$.\n\n6. The greatest integer function $\lfloor x \rfloor$ is constant on intervals between integers and jumps at integers. At $x=2$,\n\n$$\lim_{h \to 0^+} \frac{f(2+h) - f(2)}{h} = \lim_{h \to 0^+} \frac{2 - 2}{h} = 0$$\n\n$$\lim_{h \to 0^-} \frac{f(2+h) - f(2)}{h} = \lim_{h \to 0^-} \frac{1 - 2}{h} = \lim_{h \to 0^-} \frac{-1}{h} = +\infty$$\n\nSince the left-hand and right-hand limits of the difference quotient are not equal, $f$ is not differentiable at $x=2$.\n\n---\n\n7. Problem 24: Given $\vec{a} = 2 \hat{i} + 2 \hat{j} + 3 \hat{k}$, $\vec{b} = - \hat{i} + 2 \hat{j} + \hat{k}$, $\vec{c} = 3 \hat{i} + \hat{j}$, find $\lambda$ such that $(\vec{a} + \lambda \vec{b}) \perp \vec{c}$.\n\n8. Two vectors are perpendicular if their dot product is zero:\n\n$$(\vec{a} + \lambda \vec{b}) \cdot \vec{c} = 0$$\n\nCalculate the dot product:\n\n$$(2 + \lambda(-1)) \cdot 3 + (2 + \lambda 2) \cdot 1 + (3 + \lambda 1) \cdot 0 = 0$$\n\nSimplify:\n\n$$3(2 - \lambda) + (2 + 2\lambda) + 0 = 0$$\n\n$$6 - 3\lambda + 2 + 2\lambda = 0$$\n\n$$8 - \lambda = 0 \implies \lambda = 8$$\n\nFinal answers:\n\n$$\boxed{\frac{dy}{dx} = (\sin x)^x \left( \ln(\sin x) + x \cot x \right) + \frac{1}{2 \sqrt{x} \sqrt{1 - x}}}$$\n$$\boxed{f(x) = \lfloor x \rfloor \text{ is not differentiable at } x=2}$$\n$$\boxed{\lambda = 8}$$