Derivative of $\tanh\left(x\right)$
Differentiated with respect to x. Edit the function · Differentiation rules
Step-by-step solution
Input function
$$\tanh\left(x\right)$$
Start of the differentiation
$$\frac{d}{dx}{\left[\tanh\left(x\right)\right]}$$
Yellow: the expression being replaced. Green: its replacement.
Step 1 — Derivative of the hyperbolic tangent
Differentiation rule
📖 Rule
$$\frac{d}{dx}\tanh\left(x\right)=\frac{1}{\cosh\left(x\right)^{2}}$$
Why: The argument is exactly x. Apply the known elementary derivative.
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\tanh\left(x\right)\right]}\,}}}$$
✨ After: Derivative of the hyperbolic tangent
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\cosh\left(x\right)^{2}}\,}}}$$
Final result
$$\frac{1}{\cosh\left(x\right)^{2}}$$
Cross-check with a computer algebra system
The derivative computed directly by the computer algebra system Maxima:
$$\operatorname{sech}\left(x\right)^{2}$$
Step-by-step result ↔ Maxima: identical ✓