Hyperbolic Derivative Notes

Published on: September 29, 2026

This Hyperbolic Derivative Calculator differentiates sinh, cosh, tanh and their relatives, and shows the working line by line. These mirror the trigonometric derivatives closely, except that several of the minus signs are absent, which is the difference worth watching. A constant multiple in front is carried through unchanged. Following the same pattern every time keeps the hyperbolic signs from being copied from the trigonometric ones.

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that hyperbolic function.
  3. Apply the rule, keeping the constant multiple in front.

Formula:

\(\begin{gathered} \frac{d}{dx}\left(\sinh\left(x\right)\right) = \cosh\left(x\right) \\ \frac{d}{dx}\left(\cosh\left(x\right)\right) = \sinh\left(x\right) \\ \frac{d}{dx}\left(\tanh\left(x\right)\right) = \operatorname{sech}^{2}\left(x\right) \end{gathered}\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that hyperbolic function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Set up the problem.

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

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

Step 2

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

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

\(f'\left(x\right) = \cosh{\left(x \right)}\)
See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that hyperbolic function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Set up the problem.

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

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

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

\(f'\left(x\right) = 2 \operatorname{sech}^{2}{\left(x \right)}\)
See Example 3 Hide Example 3

Problem

\(f\left(x\right) = 4 \cosh{\left(x \right)}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that hyperbolic function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Set up the problem.

In this problem: The function is \(\cosh{\left(x \right)}\). The constant multiple \(4\) stays in front.

\(f\left(x\right) = 4 \cosh{\left(x \right)}\)

Step 2

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

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

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

Step 3

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

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

\(f'\left(x\right) = 4 \cdot \left(\sinh{\left(x \right)}\right) = 4 \sinh{\left(x \right)}\)

Final Answer

\(f'\left(x\right) = 4 \sinh{\left(x \right)}\)
See Example 4 Hide Example 4

Problem

\(f\left(x\right) = 3 \tanh{\left(x \right)}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that hyperbolic function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Set up the problem.

In this problem: The function is \(\tanh{\left(x \right)}\). The constant multiple \(3\) stays in front.

\(f\left(x\right) = 3 \tanh{\left(x \right)}\)

Step 2

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

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

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

\(f'\left(x\right) = 3 \cdot \left(\operatorname{sech}^{2}{\left(x \right)}\right) = 3 \operatorname{sech}^{2}{\left(x \right)}\)

Final Answer

\(f'\left(x\right) = 3 \operatorname{sech}^{2}{\left(x \right)}\)