Difference between revisions of "Derivative of sech"

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(Proof)
 
Line 9: Line 9:
 
Using the [[quotient rule]], the [[derivative of cosh]], and the definition of $\mathrm{tanh}$, we see
 
Using the [[quotient rule]], the [[derivative of cosh]], and the definition of $\mathrm{tanh}$, we see
 
$$\begin{array}{ll}
 
$$\begin{array}{ll}
\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{sech}(z) &= \dfrac{-\sinh(z)}{\cosh(z)^2} \\
+
\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{sech}(z) &= \dfrac{0-\sinh(z)}{\cosh(z)^2} \\
 
&=-\mathrm{sech}(z)\mathrm{tanh}(z),
 
&=-\mathrm{sech}(z)\mathrm{tanh}(z),
 
\end{array}$$
 
\end{array}$$

Latest revision as of 11:47, 17 September 2016

Theorem

The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{sech}(z)=-\mathrm{sech}(z)\mathrm{tanh}(z),$$ where $\mathrm{sech}$ denotes the hyperbolic secant and $\mathrm{tanh}$ denotes the hyperbolic tangent.

Proof

From the definition, $$\mathrm{sech}(z) = \dfrac{1}{\mathrm{cosh}(z)}.$$ Using the quotient rule, the derivative of cosh, and the definition of $\mathrm{tanh}$, we see $$\begin{array}{ll} \dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{sech}(z) &= \dfrac{0-\sinh(z)}{\cosh(z)^2} \\ &=-\mathrm{sech}(z)\mathrm{tanh}(z), \end{array}$$ as was to be shown.

References