How to Find the Derivative of the Hyperbolic Tangent Function (tanh(x))

d/dx (tanh x)

To find the derivative of the hyperbolic tangent function, tanh(x), with respect to x, we can use the chain rule

To find the derivative of the hyperbolic tangent function, tanh(x), with respect to x, we can use the chain rule. Let’s break down the process step by step:

Step 1: Recall the definition of the hyperbolic tangent function, tanh(x), which is given by:

tanh(x) = (e^(2x) – 1) / (e^(2x) + 1)

Step 2: Rewrite tanh(x) using an exponential form:

tanh(x) = (e^(2x) – 1) / (e^(2x) + 1) = (e^(2x) – 1) * (e^(-2x) / (e^(2x) + 1))

Step 3: Simplify the expression:

tanh(x) = (e^(2x) – e^(-2x)) / (e^(2x) + 1)

Step 4: Now, let’s find the derivative of tanh(x) using the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x) / h(x), then its derivative is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2

Applying the quotient rule, let:

g(x) = e^(2x) – e^(-2x)
h(x) = e^(2x) + 1

g'(x) = (d/dx)(e^(2x) – e^(-2x))
= 2e^(2x) + 2e^(-2x) (Using the derivative of the exponential function)
= 2(e^(2x) + e^(-2x))

h'(x) = (d/dx)(e^(2x) + 1)
= 2e^(2x) (Using the derivative of the exponential function)

Now, using the quotient rule formula, we have:

(tanh(x))’ = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
= (2(e^(2x) + e^(-2x)) * (e^(2x) + 1) – (e^(2x) – e^(-2x)) * 2e^(2x)) / (e^(2x) + 1)^2
= (2(e^(4x) + 2 + 1 – e^(2x)) – (e^(4x) – 1)) / (e^(2x) + 1)^2
= (e^(4x) + 2 – 2e^(2x) – e^(4x) + 1) / (e^(2x) + 1)^2
= (3 – 2e^(2x)) / (e^(2x) + 1)^2

Therefore, the derivative of the hyperbolic tangent function, d/dx (tanh(x)), is:

d/dx (tanh(x)) = (3 – 2e^(2x)) / (e^(2x) + 1)^2

More Answers:
Understanding the Concept of Continuity in Mathematics | Explained and Illustrated.
The Intermediate Value Theorem (IVT) | Understanding Its Applications and Importance in Continuous Functions
Mastering the Evaluation of Limits in Mathematics | Four Common Methods Explored

Error 403 The request cannot be completed because you have exceeded your quota. : quotaExceeded

Share:

Recent Posts

Don't Miss Out! Sign Up Now!

Sign up now to get started for free!