Derivative of Tanh

We now understand the derivative of tanh x. It is commonly represented as d/dx (tanh x) or (tanh x)', and its value is sech²x. The function tanh x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Hyperbolic Tangent Function: (tanh(x) = sinh(x)/cosh(x)). Quotient Rule: Rule for differentiating tanh(x) (since it consists of sinh(x)/cosh(x)). Hyperbolic Secant Function: sech(x) = 1/cosh(x).

The derivative of tanh x can be denoted as d/dx (tanh x) or (tanh x)'. The formula we use to differentiate tanh x is: d/dx (tanh x) = sech² x (or) (tanh x)' = sech² x The formula applies to all x where cosh(x) ≠ 0

We can derive the derivative of tanh 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 Product Rule We will now demonstrate that the differentiation of tanh x results in sech²x using the above-mentioned methods: By First Principle The derivative of tanh x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of tanh x using the first principle, we will consider f(x) = tanh 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) = tanh x, we write f(x + h) = tanh (x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [tanh(x + h) - tanh x] / h = limₕ→₀ [ [sinh (x + h) / cosh (x + h)] - [sinh x / cosh x] ] / h = limₕ→₀ [ [sinh (x + h ) cosh x - cosh (x + h) sinh x] / [cosh x · cosh(x + h)] ]/ h We now use the formula sinh A cosh B - cosh A sinh B = sinh (A - B). f'(x) = limₕ→₀ [ sinh (x + h - x) ] / [ h cosh x · cosh(x + h)] = limₕ→₀ [ sinh h ] / [ h cosh x · cosh(x + h)] = limₕ→₀ (sinh h)/ h · limₕ→₀ 1 / [cosh x · cosh(x + h)] Using limit formulas, limₕ→₀ (sinh h)/ h = 1. f'(x) = 1 [ 1 / (cosh x · cosh(x + 0))] = 1/cosh² x As the reciprocal of hyperbolic cosine is hyperbolic secant, we have, f'(x) = sech² x. Hence, proved. Using Chain Rule To prove the differentiation of tanh x using the chain rule, We use the formula: Tanh x = sinh x/ cosh x Consider f(x) = sinh x and g(x) = cosh x So we get, tanh 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) = sinh x and g(x) = cosh x in equation (1), d/dx (tanh x) = [(cosh x) (cosh x) - (sinh x) (-sinh x)] / (cosh x)² (cosh² x + sinh² x)/ cosh² x …(2) Here, we use the formula: (cosh² x) - (sinh² x) = 1 (Hyperbolic identity) Substituting this into (2), d/dx (tanh x) = 1 / (cosh x)² Since sech x = 1/cosh x, we write: d/dx(tanh x) = sech² x Using Product Rule We will now prove the derivative of tanh x using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, Tanh x = sinh x/ cosh x tanh x = (sinh x). (cosh x)⁻¹ Given that, u = sinh x and v = (cosh x)⁻¹ Using the product rule formula: d/dx [u.v] = u'. v + u. v' u' = d/dx (sinh x) = cosh x. (substitute u = sinh x) Here we use the chain rule: v = (cosh x)⁻¹ = (cosh x)⁻¹ (substitute v = (cosh x)⁻¹) v' = -1. (cosh x)⁻² . d/dx (cosh x) v' = -sinh x / (cosh x)² Again, use the product rule formula: d/dx (tanh x) = u'. v + u. v' Let’s substitute u = sinh x, u' = cosh x, v = (cosh x)⁻¹, and v' = -sinh x / (cosh x)² When we simplify each term: We get, d/dx (tanh x) = 1 - sinh²x / (cosh x)² sinh² x/ (cosh x)² = tanh² x (we use the identity sinh² x = cosh² x - 1) Thus: d/dx (tanh x) = 1 - tanh² x Since, 1 - tanh² x = sech² x d/dx (tanh x) = sech² 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 tanh(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 tanh(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, as tanh(x) approaches 1, and thus its derivative sech²(x) approaches zero. When x is 0, the derivative of tanh x = sech²(0), which is 1.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Hyperbolic Tangent Function: The hyperbolic tangent function is a hyperbolic function represented as tanh x. Hyperbolic Secant Function: A hyperbolic function that is the reciprocal of the hyperbolic cosine function. It is typically represented as sech x. First Derivative: It is the initial result of a function, which gives us the rate of change of a specific function. Asymptote: The function approaches a line without intersecting or crossing it. This line is known as an asymptote.