Derivative of tanh(x)
Answer
Derivative of tanh(x)
- Rule Used
- Hyperbolic Rule
- Steps
- Recognize a hyperbolic function. sinh and cosh swap under differentiation, with no minus sign, because both are built from e^x and with the same sign pattern.
- Final answer: sech(x)^2
d/dx tanh(x) = sech(x)^2. The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.
Instant step-by-step solution for tanh(x)
Derivative of tanh(x) = sech(x)^2 Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-tanh-x
Derivative
tanh(x) → sech(x)^2engine: sech(x)^2
Graph of the function and its derivative
The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.
Why the derivative has this form
tanh(x) = sinh/cosh. The quotient rule produces cosh^2 − sinh^2 in the numerator, which equals 1, over cosh^2. That is sech^2(x). It is the hyperbolic analogue of sec^2, and the sign stays positive because cosh^2 − sinh^2 = 1 rather than the minus that showed up for cotangent.
Apply the exponential rule to tanh(x). The form to remember, already simplified, is sech(x)^2.
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.
At x = 0, substitute into the simplified form sech(x)^2. That number is the slope of tanh(x) there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = tanh(x).
- Apply the exponential rule to tanh(x). The form to remember, already simplified, is sech(x)^2.
- At x = 0, substitute into the simplified form sech(x)^2. That number is the slope of tanh(x) there; a secant over a small step in x lands next to it.
Common mistakes
Answering -sech(x)^2
A leading minus copies cotangent's derivative onto tanh. The identity for cosh and sinh is a difference equal to +1, so the numerator does not change sign.
Copying a trigonometric minus onto tanh(x)
Hyperbolic sine and cosine come from exponentials. The minus signs inside those definitions do not leave a minus in front of the derivative the way cosine does.
Related rule and examples
- Exponential derivatives — The rule page lists this example and the others in the same family.
- Quotient rule — The rule page lists this example and the others in the same family.
- Derivatives of trigonometric functions — Same family of differentiation, different function.
- Derivative of sinh(x) — Same family of differentiation, different function.
- Derivative of cosh(x) — Same family of differentiation, different function.
Frequently asked questions
What is the derivative of tanh(x)?
sech(x)^2. tanh(x) = sinh/cosh. The quotient rule produces cosh^2 − sinh^2 in the numerator, which equals 1, over cosh^2. That is sech^2(x). It is the hyperbolic analogue of sec^2, and the sign stays positive because cosh^2 − sinh^2 = 1 rather than the minus that showed up for cotangent.
Which wrong answer shows up for tanh(x)?
A leading minus copies cotangent's derivative onto tanh. The identity for cosh and sinh is a difference equal to +1, so the numerator does not change sign.
Where is the derivative of tanh(x) defined?
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.