\( y=\sinh (x) \)

y = sinh(x) is the hyperbolic sine function. It is defined using the exponential function as follows:   sinh(x) = (e^x – e^(–x)) / 2 Here are some key properties and facts about sinh(x): 1. Definition and Expression:   • sinh(x) = (e^x – e^(–x)) / 2   This definition expresses sinh(x) as the difference of two exponential functions divided by 2. 2. Derivative:   • The derivative of sinh(x) is cosh(x), where:     cosh(x) = (e^x + e^(–x)) / 2   Thus, d/dx [sinh(x)] = cosh(x). 3. Integral:   • The integral of sinh(x) is cosh(x) plus a constant:     ∫ sinh(x) dx = cosh(x) + C. 4. Series Expansion:   • The Maclaurin series expansion for sinh(x) is:     sinh(x) = x + (x^3)/3! + (x^5)/5! + (x^7)/7! + …   This infinite series converges for all x ∈ ℝ. 5. Inverse Function:   • The inverse hyperbolic sine function is denoted as arsinh(x) or sometimes as sinh⁻¹(x). Its expression is given by:     arcsinh(x) = ln(x + √(x² + 1)). 6. Graph Behavior:   • The function is odd: sinh(–x) = –sinh(x). This symmetry implies that its graph is symmetric with respect to the origin.   • As x → ∞, sinh(x) grows approximately as (e^x)/2, and as x → –∞ it behaves like –(e^(–x))/2.   • The graph of sinh(x) is smooth and continuous, with a characteristic S-shape due to its exponential components. 7. Applications:   • Hyperbolic functions like sinh(x) come up frequently in the solutions of differential equations, in describing the shape of a hanging cable (catenary), and in many areas of physics and engineering. These properties and characteristics make sinh(x) an important function in both pure and applied mathematics.