Derivative of Hyperbolic Sine Function

Theorem

Let $u$ be a differentiable real function of $x$.

Then:

$\map {\dfrac \d {\d x} } {\sinh u} = \cosh u \dfrac {\d u} {\d x}$

where $\sinh$ is the hyperbolic sine and $\cosh$ is the hyperbolic cosine.

Proof

\(\ds \map {\frac \d {\d x} } {\sinh u}\)

\(=\)

\(\ds \map {\frac \d {\d u} } {\sinh u} \frac {\d u} {\d x}\)

Chain Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \cosh u \frac {\d u} {\d x}\)

Derivative of Hyperbolic Sine

$\blacksquare$

Also see

• Derivative of Hyperbolic Cosine Function

• Derivative of Hyperbolic Tangent Function
• Derivative of Hyperbolic Cotangent Function

• Derivative of Hyperbolic Secant Function
• Derivative of Hyperbolic Cosecant Function

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Derivatives of Hyperbolic and Inverse Hyperbolic Functions: $13.31$
• 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $15$: Differentiation of Hyperbolic Functions: Differentiation Formulas: $31$.
• 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $4$. Derivatives: Derivatives of Special Functions: $19$