Question

Verify the differentiation formula. \(\frac{d}{d x}[\operatorname{sech} x]=-\operatorname{sech} x \tanh x\)

Short answer

The differentiation formula \(\frac{d}{d x}[sech x] = -sech(x) tanh(x)\) has been successfully verified.

Step-by-step solution

Definition of sech

Firstly, define the hyperbolic secant function. Recall that \(sech(x) = \frac{2}{e^{x} + e^{-x}}\). Then, rewrite the function in terms of exponential functions.

Differentiate sech x using Chain Rule

Next, differentiate \(sech(x)\) using chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. So differentiate \(sech(x) = \frac{2}{e^{x} + e^{-x}}\), we get that \(\frac{d}{d x}[sech x] = - sech(x) (e^x - e^{-x})\).

Simplify and get tanh x

Now, simplify the right side of the expression. Recall that \(- (e^x - e^{-x})\ = tanh(x)\). Thus, we can conclude that \(- sech(x) (e^x - e^{-x}) = - sech(x) tanh(x)\).

Verify the formula

Now that the expression is simplified to \(- sech(x) tanh(x)\), it matches with the given formula \(\frac{d}{d x}[sech x] = -sech(x) tanh(x)\), and so we can say that the differentiation formula has been successfully verified.