How can I find the derivative $ \operatorname{sech}^2(e^x)$?

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Please can anyone help me to take the derivative of $\; \operatorname{sech}^2(e^x)$?

Thanks all

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2 Answers

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Use the chain rule, since we have a composition of three functions:

$$f(x) = x^2, \quad g(x) = \operatorname{sech}(x), \quad h(x) = e^x$$

Then $$f\circ g \circ h = f(g(h(x))) = \left(\operatorname{sech}(e^x)\right)^2 = \operatorname{sech}^2(e^x)$$

And $$\frac{d}{dx} \Big((f\circ g\circ h)(x)\Big) = f'(g(h(x)))\cdot g'(h(x)) \cdot h'(x)$$

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$$\begin{align}\frac d{dx} \Big(\operatorname{sech}^2(e^x)\Big) = \frac{d}{dx}\left(\operatorname{sech}(e^x)\right)^2 & = \underbrace{2\left(\operatorname{sech}(e^x)\right)}_{\large f'(g(h(x)))}\cdot\underbrace{\left(-\operatorname{tanh}(e^x)\operatorname{sech}(e^x)\right)}_{\large g'(h(x))}\cdot \underbrace{e^x}_{\large h'(x)}\\ \\ \\ \\& = -2e^x\operatorname{sech}^2(e^x) \operatorname{tanh}(e^x)\end{align}$$

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Hint: For starters write it as

$$y=(\text{sech}(e^x))^2.$$

Do you know the Chain Rule? The derivative of $\text{sech } x$?

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