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MathCalculus

Hyperbolic Sine: Differentiate sinh(e^x) and Check the Domain

Differentiate the hyperbolic sine composite sinh(e^x), keep its exponential argument, and verify the derivative and its real domain.

Question

Differentiate the hyperbolic sine function f(x)=sinh⁡(ex)f(x)=\sinh(e^x) with respect to xx, and state the real domain of the function and of its derivative.

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Evidence boundary

This page solves the independently selected Exercise 9(a) from the University of Melbourne bridging notes, including its domain instruction. It does not claim to solve the other parts of Exercise 9. The derivative is derived here and agrees with the printed answer for this item.