1. **Stating the problem:** We are given the function $$y(x) = 3\sqrt{x} + 4\cosh x - \frac{2}{x^3} + 5\ln x$$ and we want to understand it clearly step by step. 2. **Understanding each term:** - The first term is $$3\sqrt{x}$$ which means 3 times the square root of $$x$$. - The second term is $$4\cosh x$$ where $$\cosh x$$ is the hyperbolic cosine function defined as $$\cosh x = \frac{e^x + e^{-x}}{2}$$. - The third term is $$- \frac{2}{x^3}$$ which is negative 2 divided by $$x$$ cubed. - The fourth term is $$5\ln x$$ which is 5 times the natural logarithm of $$x$$. 3. **Important domain considerations:** - $$\sqrt{x}$$ requires $$x \geq 0$$. - $$\ln x$$ requires $$x > 0$$. - $$\frac{2}{x^3}$$ requires $$x \neq 0$$. Combining these, the domain of $$y(x)$$ is $$x > 0$$. 4. **Formula for derivative (if needed):** - Derivative of $$\sqrt{x}$$ is $$\frac{1}{2\sqrt{x}}$$. - Derivative of $$\cosh x$$ is $$\sinh x$$. - Derivative of $$\frac{1}{x^3}$$ is $$-3x^{-4}$$. - Derivative of $$\ln x$$ is $$\frac{1}{x}$$. 5. **Simplifying or evaluating:** If you want to evaluate $$y(x)$$ at a specific $$x$$, substitute the value and compute each term. This explanation breaks down the function into understandable parts and clarifies the domain and components.