Example 1 Find the derivatives of a) $$ anh x-\frac{1}{3} anh ^{3} x $$

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Example 1 Find the derivatives of a) $$ 	anh x-\frac{1}{3} 	anh ^{3} x $$

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To find the derivative of the function $$ f(x) = \tanh x - \frac{1}{3} \tanh^3 x $$, we will use the chain rule and the derivative of the hyperbolic tangent function.

The derivative of $$ \tanh x $$ is given by:
$$
\frac{d}{dx}(\tanh x) = \text{sech}^2 x
$$

Now, we can differentiate $$ f(x) $$:

1. Differentiate $$ \tanh x $$:
   $$
   \frac{d}{dx}(\tanh x) = \text{sech}^2 x
   $$

2. Differentiate $$ -\frac{1}{3} \tanh^3 x $$ using the chain rule:
   $$
   \frac{d}{dx}\left(-\frac{1}{3} \tanh^3 x\right) = -\frac{1}{3} \cdot 3 \tanh^2 x \cdot \frac{d}{dx}(\tanh x) = -\tanh^2 x \cdot \text{sech}^2 x
   $$

Now, combine the derivatives:
$$
f'(x) = \text{sech}^2 x - \tanh^2 x \cdot \text{sech}^2 x
$$

Factor out $$ \text{sech}^2 x $$:
$$
f'(x) = \text{sech}^2 x \left(1 - \tanh^2 x\right)
$$

Using the identity $$ 1 - \tanh^2 x = \text{sech}^2 x $$, we can simplify further:
$$
f'(x) = \text{sech}^2 x \cdot \text{sech}^2 x = \text{sech}^4 x
$$

Thus, the derivative of the function $$ f(x) = \tanh x - \frac{1}{3} \tanh^3 x $$ is:
$$
f'(x) = \text{sech}^4 x
$$
The derivative of $$ \tanh x - \frac{1}{3} \tanh^3 x $$ is $$ \text{sech}^4 x $$.

Example 1
Find the derivatives of
a)

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