Derivative of Hyperbolic Cosecant using First Principle of Derivatives

The derivative of

\text{csch}(x)

-\text{csch}(x)\text{coth}(x)

using first principle of derivatives.

Proof. Let

f(x) = \text{csch}(x) = \frac{2}{e^x - e^{-x}}

. Then we use the first principle of derivatives to determine

\text{sech}(x)

\begin{align*} f'(x) &= \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h} \\ &= \lim_{h \rightarrow 0} \frac{\text{csch}(x + h) - \text{csch}(x)}{h} \\ &= \lim_{h \rightarrow 0} \frac{\frac{2}{e^{x + h} - e^{-x - h}} - \frac{2}{e^x - e^{-x}}}{h} \\ &= \lim_{h \rightarrow 0} \frac{\frac{2e^{x} - 2e^{-x} - 2e^{x + h} + 2e^{-x - h}}{(e^{x + h} - e^{-x - h})(e^{x} - e^{-x})}}{h} \\ &= 2 \cdot \lim_{h \rightarrow 0} \frac{\frac{e^{x} - e^{-x} - e^{x + h} + e^{-x - h}}{(e^{x + h} - e^{-x - h})(e^{x} - e^{-x})}}{h} \\ &= \frac{2}{e^{x} - e^{-x}} \cdot \lim_{h \rightarrow 0} \frac{\frac{e^{x} - e^{-x} - e^{x + h} + e^{-x - h}}{e^{x + h} - e^{-x - h}}}{h} \\ &= \text{csch}(x) \cdot \lim_{h \rightarrow 0} \frac{\frac{e^{x} - e^{-x} - e^{x + h} + e^{-x - h}}{e^{x + h} - e^{-x - h}}}{h} \\ &= \text{csch}(x) \cdot \lim_{h \rightarrow 0} \frac{e^{x} - e^{-x} - e^{x + h} + e^{-x - h}}{h(e^{x + h} - e^{-x - h})} \\ &= \text{csch}(x) \cdot \lim_{h \rightarrow 0} \frac{1}{e^{x + h} - e^{-x - h}} \cdot \lim_{h \rightarrow 0} \frac{e^{x} - e^{-x} - e^{x + h} + e^{-x - h}}{h} \\ &= \text{csch}(x) \cdot \frac{1}{e^{x} - e^{-x}} \cdot \lim_{h \rightarrow 0} \frac{e^{x} - e^{-x} - e^{x + h} + e^{-x - h}}{h} \\ &= \text{csch}(x) \cdot \frac{1}{e^{x} - e^{-x}} \cdot \lim_{h \rightarrow 0} \frac{e^{x}(1 - e^{h}) - e^{-x}(1 - e^{-h})}{h} \\ &= -\text{csch}(x) \cdot \frac{1}{e^{x} - e^{-x}} \cdot \lim_{h \rightarrow 0} \frac{e^{x}(e^{h} - 1) - e^{-x}(e^{-h} - 1)}{h}. \\ \end{align*}

We want to determine the RHS of the above equation:

\begin{align*} \lim_{h \rightarrow 0} \frac{e^{x}(e^{h} - 1) - e^{-x}(e^{-h} - 1)}{h} &= \lim_{h \rightarrow 0} \frac{e^{x}(e^{h} - 1)}{h} - \lim_{h \rightarrow 0} \frac{e^{-x}(e^{-h} - 1)}{h} \\ &= e^{x} \cdot \lim_{h \rightarrow 0} \frac{(e^{h} - 1)}{h} - e^{-x} \cdot \lim_{h \rightarrow 0} \frac{(e^{-h} - 1)}{h} \\ \end{align*}

We know from here that

\lim_{h \rightarrow 0} \frac{(e^{h} - 1)}{h} = 1

and

\lim_{h \rightarrow 0} \frac{(e^{-h} - 1)}{h} = -1

. So we get

\begin{align*} \lim_{h \rightarrow 0} \frac{e^{x}(e^{h} - 1) - e^{-x}(e^{-h} - 1)}{h} = e^{x} + e^{-x}. \end{align*}

Putting everything together, we get:

\begin{align*} f'(x) &= -\text{csch}(x) \cdot \frac{1}{e^{x} - e^{-x}} \cdot (e^{x} + e^{-x}) \\ &= -\text{csch}(x) \cdot \frac{e^{x} + e^{-x}}{e^{x} - e^{-x}} \\ &= -\text{csch}(x) \text{coth}(x). \end{align*}

So, the derivative of

\text{csch}(x)

-\text{csch}(x)\text{coth}(x)

using first principle of derivatives.