1. Differentiate $y = x \sin x$ with respect to $x$. Use the product rule: $\frac{d}{dx}[uv] = u'v + uv'$ where $u = x$ and $v = \sin x$. Calculate derivatives: $u' = 1$, $v' = \cos x$. Apply the rule: $$\frac{dy}{dx} = 1 \cdot \sin x + x \cdot \cos x = \sin x + x \cos x.$$ 2. Differentiate $y = x^2 e^{2x}$ with respect to $x$. Use the product rule with $u = x^2$ and $v = e^{2x}$. Calculate derivatives: $u' = 2x$, $v' = 2 e^{2x}$ (chain rule). Apply the rule: $$\frac{dy}{dx} = 2x e^{2x} + x^2 (2 e^{2x}) = 2x e^{2x} + 2x^2 e^{2x} = 2x e^{2x}(1 + x).$$ 3. Differentiate $y = x^2 \ln x$ with respect to $x$. Use the product rule with $u = x^2$ and $v = \ln x$. Calculate derivatives: $u' = 2x$, $v' = \frac{1}{x}$. Apply the rule: $$\frac{dy}{dx} = 2x \ln x + x^2 \cdot \frac{1}{x} = 2x \ln x + x.$$ 4. Differentiate $y = \sec(ax)$ with respect to $x$. Recall derivative: $\frac{d}{dx} \sec u = \sec u \tan u \cdot u'$. Here, $u = ax$, so $u' = a$. Apply the chain rule: $$\frac{dy}{dx} = \sec(ax) \tan(ax) \cdot a = a \sec(ax) \tan(ax).$$ 5. Differentiate $y = \frac{2 \cos 3x}{x^3}$ with respect to $x$. Use the quotient rule: $\frac{d}{dx} \left( \frac{f}{g} \right) = \frac{f'g - fg'}{g^2}$ where $f = 2 \cos 3x$, $g = x^3$. Calculate derivatives: $$f' = 2 \cdot (-\sin 3x) \cdot 3 = -6 \sin 3x,$$ $$g' = 3x^2.$$ Apply the quotient rule: $$\frac{dy}{dx} = \frac{(-6 \sin 3x) x^3 - 2 \cos 3x (3x^2)}{x^6} = \frac{-6 x^3 \sin 3x - 6 x^2 \cos 3x}{x^6}.$$ Simplify numerator: $$= \frac{-6 x^2 (x \sin 3x + \cos 3x)}{x^6} = -6 \frac{x \sin 3x + \cos 3x}{x^4}.$$ 6. Find the gradient of $y = \frac{2x}{x^2 - 5}$ at the point $(2, -4)$. Use the quotient rule with $f = 2x$, $g = x^2 - 5$. Calculate derivatives: $$f' = 2,$$ $$g' = 2x.$$ Apply quotient rule: $$\frac{dy}{dx} = \frac{2 (x^2 - 5) - 2x (2x)}{(x^2 - 5)^2} = \frac{2x^2 - 10 - 4x^2}{(x^2 - 5)^2} = \frac{-2x^2 - 10}{(x^2 - 5)^2}.$$ Evaluate at $x=2$: $$\frac{dy}{dx}\bigg|_{x=2} = \frac{-2(2)^2 - 10}{(2^2 - 5)^2} = \frac{-8 - 10}{(4 - 5)^2} = \frac{-18}{(-1)^2} = -18.$$ Final answers: 1. $\frac{dy}{dx} = \sin x + x \cos x$ 2. $\frac{dy}{dx} = 2x e^{2x} (1 + x)$ 3. $\frac{dy}{dx} = 2x \ln x + x$ 4. $\frac{dy}{dx} = a \sec(ax) \tan(ax)$ 5. $\frac{dy}{dx} = -6 \frac{x \sin 3x + \cos 3x}{x^4}$ 6. Gradient at $(2, -4)$ is $-18$.