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e^{-x}}{2}\n$\n- **Derivatives of Hyperbolic Functions:**  \n$\n\\frac{d}{dx} \\cosh(x) = \\sinh(x), \\quad \\frac{d}{dx} \\sinh(x) = \\cosh(x)\n$\n- **Identity for $$ \\cosh^2(x) - \\sinh^2(x) = 1 $$**\n\n---\n\n## **\ud83d\udd35 Problem 377: Expressions for $$ \\cosh x + \\sinh x $$ and $$ \\cosh x - \\sinh x $$**\n\u2714 **Step 1: Use Definitions**\n$\n\\cosh(x) + \\sinh(x) = \\frac{e^x + e^{-x}}{2} + \\frac{e^x - e^{-x}}{2} = e^x\n$\n$\n\\cosh(x) - \\sinh(x) = \\frac{e^x + e^{-x}}{2} - \\frac{e^x - e^{-x}}{2} = e^{-x}\n$\n\n\ud83d\udccc **Final Answer:**  \n\u2714 **$$ \\cosh x + \\sinh x = e^x $$**  \n\u2714 **$$ \\cosh x - \\sinh x = e^{-x} $$**\n\n---\n\n## **\ud83d\udd35 Problem 380: Verifying $$ \\tanh(x)' = \\text{sech}^2(x) $$**\n\u2714 **Step 1: Use the Definition**\n$\n\\tanh(x) = \\frac{\\sinh(x)}{\\cosh(x)}\n$\n\u2714 **Step 2: Apply Quotient Rule**\n$\n\\frac{d}{dx} \\left( \\frac{\\sinh x}{\\cosh x} \\right) = \\frac{ \\cosh x \\cdot \\cosh x - \\sinh x \\cdot \\sinh x }{ \\cosh^2 x }\n$\n\u2714 **Step 3: Use Identity**\n$\n\\cosh^2 x - 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