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The derivative formulas are as follows: 1. Trigonometric functions: - Derivative of sine function: \(\frac{d}{dx}(\sin(x)) = \cos(x)\) - Derivative of cosine function: \(\frac{d}{dx}(\cos(x)) = -\sin(x)\) 2. Hyperbolic functions: - Derivative of hyperbolic sine function: \(\frac{d}{dx}(\sinh(x)) = \cosh(x)\) - Derivative of hyperbolic cosine function: \(\frac{d}{dx}(\cosh(x)) = \sinh(x)\) Now, we will compare these derivative formulas to identify the similarities and differences.

In both cases (trigonometric and hyperbolic functions), we can observe that the derivative of one function gives us the other function. Specifically, we have: - For trigonometric functions: The derivative of the sine function gives us the cosine function, and the derivative of the cosine function gives us the sine function (with a negative sign). - For hyperbolic functions: The derivative of the hyperbolic sine function gives us the hyperbolic cosine function, and the derivative of the hyperbolic cosine function gives us the hyperbolic sine function. As both the trigonometric and hyperbolic functions are periodic in nature, the derivatives are also related to functions of the same family.

On the other hand, the differences in the derivative formulas can be highlighted as follows: - The derivative of the trigonometric cosine function has a negative sign, whereas the derivative of the hyperbolic cosine function remains positive. - The trigonometric functions deal with angles while the hyperbolic functions - with exponential notions. As such, their geometric meanings and interpretations differ significantly. In conclusion, the derivative formulas for the hyperbolic functions and the trigonometric functions have some similarities in the sense that the derivative of one function gives us the other function from the same family. However, they also have differences in terms of the signs that accompany the resulting derivatives and the concepts they represent in mathematics.