Derivatives of Hyperbolic Functions — Question 5

Question 5

Let $f(x) = \frac{\sinh(x)}{\cosh(x)}$.

• Simplify the function if possible.

• Find $f'(x)$.

• Determine the intervals where $f(x)$ is increasing or decreasing.

Question 5 - Solution

(a) Simplify the Function:

We have: $$f(x) = \frac{\sinh(x)}{\cosh(x)}$$

This is the definition of the hyperbolic tangent function: $$f(x) = \tanh(x)$$

(b) Differentiate $f(x) = \tanh(x)$:

The derivative of $\tanh(x)$ is: $$f'(x) = \text{sech}^2(x)$$

(c) Increasing/Decreasing Intervals:

Since $f'(x) = \text{sech}^2(x) > 0$ for all real $x$, the function is strictly increasing on its entire domain.

Conclusion: $$\boxed{f'(x) = \text{sech}^2(x), \quad f(x) \text{ is increasing on } (-\infty, \infty)}$$