Chapter 2: Differentiation

Section 2.9: The Hyperbolic Functions and Their Derivatives

Example 2.9.2

$$ Verify the differentiation rule for $\tanh \left(x\right)$. $$ Solution $$ Similar to the approach taken in Section 2.6 for trig functions, express $\tanh \left(x\right)$ in terms of the hyperbolic sine and cosine; then differentiate. The details are as follows.   $\frac{d}{\mathrm{dx}}\tanh \left(x\right)$ $\=\frac{d}{\mathrm{dx}}\left(\frac{\sinh \left(x\right)}{\cosh \left(x\right)}\right)$ $$ $\=\frac{\cosh \left(x\right)\cdot \cosh \left(x\right)-\sinh \left(x\right)\cdot \sinh \left(x\right)}{{\cosh }^2\left(x\right)}$ $$ $\=\frac{{\cosh }^2\left(x\right)-{\sinh }^2\left(x\right)}{{\cosh }^2\left(x\right)}$ $$ $\=\frac{1}{{\cosh }^2\left(x\right)}$ $$ $\={\mathrm{sech}}^2\left(x\right)$   Note the use of the quotient rule to effect the differentiation on the right-hand side of the first equality.

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