Derivative of sech x

We now understand the derivative of sech x. It is commonly represented as d/dx (sech x) or (sech x)', and its value is -sech(x)tanh(x). The function sech x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Hyperbolic Secant Function: sech(x) = 1/cosh(x).

Quotient Rule: Rule for differentiating sech(x) (since it consists of 1/cosh(x)).

Hyperbolic Tangent Function: tanh(x) = sinh(x)/cosh(x).

The derivative of sech x can be denoted as d/dx (sech x) or (sech x)'. The formula we use to differentiate sech x is: d/dx (sech x) = -sech(x)tanh(x) (sech x)' = -sech(x)tanh(x) The formula applies to all x where cosh(x) ≠ 0.

We can derive the derivative of sech x using proofs. To show this, we will use the hyperbolic identities along with the rules of differentiation.

There are several methods we use to prove this, such as:

• By First Principle
   
• Using Chain Rule
   
• Using Quotient Rule

We will now demonstrate that the differentiation of sech x results in -sech(x)tanh(x) using the above-mentioned methods:

By First Principle

The derivative of sech x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of sech x using the first principle, we will consider f(x) = sech x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = sech x, we write f(x + h) = sech (x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [sech(x + h) - sech x] / h Using the definition of sech and trigonometric identities, we can simplify it to show that: f'(x) = -sech(x)tanh(x). Hence, proved.

Using Chain Rule

To prove the differentiation of sech x using the chain rule, We use the formula: sech x = 1/ cosh x Consider f(x) = 1 and g (x)= cosh x So we get, sech x = f(x)/g(x) By quotient rule: d/dx [f(x) / g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]²… (1) Let’s substitute f(x) = 1 and g(x) = cosh x in equation (1), d/dx (sech x) = [0(cosh x) - 1(sinh x)] / (cosh x)² = -sinh x / (cosh x)² = -sech(x)tanh(x) Since tanh x = sinh x/cosh x, we write: d/dx(sech x) = -sech(x)tanh(x).

Using Quotient Rule

We will now prove the derivative of sech x using the quotient rule. The step-by-step process is demonstrated below: Here, we use the formula, sech x = 1/cosh x Given that, u = 1 and v = cosh x Using the quotient rule formula: d/dx [u/v] = [u'v - uv'] / v² u' = d/dx (1) = 0 (substitute u = 1) v' = d/dx (cosh x) = sinh x Using the quotient rule formula: d/dx (sech x) = (0 . v - 1 . sinh x) / (cosh x)² = -sinh x / (cosh x)² Since -sinh x / (cosh x)² = -sech(x)tanh(x), we have: d/dx (sech x) = -sech(x)tanh(x).

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky. To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like sech(x).

For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues.

For the nth Derivative of sech(x), we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change (continuing for higher-order derivatives).

When x approaches infinity, the derivative approaches zero because sech(x) tends to zero.

When x is zero, the derivative of sech x = -sech(x)tanh(x), which is 0 because tanh(0)=0.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Hyperbolic Secant Function: A hyperbolic function represented as sech x, the reciprocal of the hyperbolic cosine.

• Hyperbolic Tangent Function: A hyperbolic function represented as tanh x, the ratio of sinh x to cosh x.

• Quotient Rule: A method for finding the derivative of a quotient of two functions.

• Chain Rule: A formula for computing the derivative of the composition of two or more functions.