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Use the exponential definition of hyperbolic functions to compute d/dx tanh(x)

To compute the derivative \(\frac{d}{dx} \tanh(x)\) using the exponential definition of hyperbolic functions, we'll start by expressing \(\tanh(x)\) in terms of exponential functions.

### Step 1: Express \(\tanh(x)\) using its exponential definition
\[
\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{\frac{e^x - e^{-x}}{2}}{\frac{e^x + e^{-x}}{2}} = \frac{e^x - e^{-x}}{e^x + e^{-x}}
\]

### Step 2: Differentiate \(\tanh(x)\)
To find \(\frac{d}{dx} \tanh(x)\), use the quotient rule, which states that for a function of the form \(\frac{u(x)}{v(x)}\), the derivative is:
\[
\frac{d}{dx} \left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\]
Here, \(u(x) = e^x - e^{-x}\) and \(v(x) = e^x + e^{-x}\).

First, find the derivatives of \(u(x)\) and \(v(x)\):
\[
u'(x) = \frac{d}{dx}(e^x - e^{-x}) = e^x + e^{-x}
\]
\[
v'(x) = \frac{d}{dx}(e^x + e^{-x}) = e^x - e^{-x}
\]

Now, apply the quotient rule:
\[
\frac{d}{dx} \tanh(x) = \frac{(e^x + e^{-x})(e^x + e^{-x}) - (e^x - e^{-x})(e^x - e^{-x})}{(e^x + e^{-x})^2}
\]

### Step 3: Simplify the expression
Simplify the numerator:
\[
(e^x + e^{-x})^2 - (e^x - e^{-x})^2
\]
This can be rewritten using the difference of squares:
\[
(a + b)^2 - (a - b)^2 = 4ab
\]
Here, \(a = e^x\) and \(b = e^{-x}\). So:
\[
4e^x e^{-x} = 4
\]

Thus, the derivative simplifies to:
\[
\frac{d}{dx} \tanh(x) = \frac{4}{(e^x + e^{-x})^2}
\]

Since \(\cosh^2(x) = \left(\frac{e^x + e^{-x}}{2}\right)^2\), we have:
\[
\frac{d}{dx} \tanh(x) = \frac{4}{4\cosh^2(x)} = \frac{1}{\cosh^2(x)}
\]

This is also written as:
\[
\frac{d}{dx} \tanh(x) = \text{sech}^2(x)
\]

So, the derivative of \(\tanh(x)\) is \(\text{sech}^2(x)\).

Would you like further details or have any questions?

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Here are 8 follow-up questions related to hyperbolic functions and their properties:

1. How would you compute the derivative of \(\sinh(x)\) and \(\cosh(x)\) directly?
2. What is the integral of \(\text{sech}^2(x)\)?
3. How can hyperbolic functions be used to solve certain types of differential equations?
4. What is the derivative of \(\coth(x)\) using its exponential definition?
5. How do the derivatives of trigonometric and hyperbolic functions compare?
6. What are the double angle identities for hyperbolic functions?
7. How would you express \(\text{sech}(x)\) in terms of exponential functions?
8. Can you find the second derivative of \(\tanh(x)\)?

**Tip:** When differentiating hyperbolic functions, their properties and identities are often similar to trigonometric functions, which can help in solving complex problems.

Learn how to compute the derivative d/dx tanh(x) using the exponential definition of hyperbolic functions. This problem covers differentiation techniques, hyperbolic identities, and the quotient rule, suitable for advanced undergraduate students.

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