Hyperbolic Derivative Calculator

Published on: July 18, 2026
Final Answer: Free • Full Steps: Plus

This Hyperbolic Derivative Calculator differentiates the hyperbolic functions sinh(x), cosh(x), and tanh(x). Each has its own derivative rule, and any constant multiple stays in front.

Step-by-step method

  1. Identify the hyperbolic function and any constant multiple.
  2. Write the derivative rule for that hyperbolic function.
  3. Apply the rule, keeping the constant multiple in front.

Formula:

\(\begin{gathered} \frac{d}{dx}\left(\sinh\left(x\right)\right) = \cosh\left(x\right) \\ \frac{d}{dx}\left(\cosh\left(x\right)\right) = \sinh\left(x\right) \\ \frac{d}{dx}\left(\tanh\left(x\right)\right) = \operatorname{sech}^{2}\left(x\right) \end{gathered}\)

Example 1:

\(f\left(x\right) = \sinh{\left(x \right)}\)

Step 1 - Identify the hyperbolic function and any constant multiple.

In this problem: The function is \(\sinh{\left(x \right)}\).

\(f\left(x\right) = \sinh{\left(x \right)}\)

Step 2 - Write the derivative rule for that hyperbolic function.

In this problem: The derivative of \(\sinh{\left(x \right)}\) is \(\cosh{\left(x \right)}\).

\(\frac{d}{dx}\left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}\)

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the function with its derivative and simplify.

\(f'\left(x\right) = \cosh{\left(x \right)}\)

Final answer:

\(f'\left(x\right) = \cosh{\left(x \right)}\)

Example 2:

\(f\left(x\right) = 2 \tanh{\left(x \right)}\)

Step 1 - Identify the hyperbolic function and any constant multiple.

In this problem: The function is \(\tanh{\left(x \right)}\). The constant multiple \(2\) stays in front.

\(f\left(x\right) = 2 \tanh{\left(x \right)}\)

Step 2 - Write the derivative rule for that hyperbolic function.

In this problem: The derivative of \(\tanh{\left(x \right)}\) is \(\operatorname{sech}^{2}{\left(x \right)}\).

\(\frac{d}{dx}\left(\tanh{\left(x \right)}\right) = \operatorname{sech}^{2}{\left(x \right)}\)

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the function with its derivative and simplify.

\(f'\left(x\right) = 2 \cdot \left(\operatorname{sech}^{2}{\left(x \right)}\right) = 2 \operatorname{sech}^{2}{\left(x \right)}\)

Final answer:

\(f'\left(x\right) = 2 \operatorname{sech}^{2}{\left(x \right)}\)