Derivative Calculator

Derivative of $\tanh\left(x\right)^{2}$

Differentiated with respect to x. Edit the function · Differentiation rules

Step-by-step solution

Input function
$$\tanh\left(x\right)^{2}$$
Start of the differentiation
$$\frac{d}{dx}{\left[\tanh\left(x\right)^{2}\right]}$$

Yellow: the expression being replaced. Green: its replacement.

Step 1 — Power Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}\,u'$$
With:
  • $u = \tanh\left(x\right)$
  • $c = 2$

Why: Differentiate the outer power: bring down the exponent and subtract one from it. The chain rule adds the derivative of the entire inner expression as another factor.

How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\tanh\left(x\right)^{2}\right]}\,}}}$$
✨ After: Power Rule
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\tanh\left(x\right)^{2 - 1}\,\frac{d}{dx}{\left[\tanh\left(x\right)\right]}\,}}}$$
Step 2 — Evaluate numbers Simplification
📖 Rule

Why: Evaluate only the highlighted numeric expression, preserving exact fractions and decimals.

Changed part: $2 - 1 \;\Longrightarrow\; 1$
🧮 Current expression
$$2\,\tanh\left(x\right)^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 - 1\,}}}}\,\frac{d}{dx}{\left[\tanh\left(x\right)\right]}$$
✨ After: Evaluate numbers
$$2\,\tanh\left(x\right)^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}\,\frac{d}{dx}{\left[\tanh\left(x\right)\right]}$$
Step 3 — Power with exponent one Simplification
📖 Rule
$$a^1=a$$

Why: An expression to the first power is the expression itself.

Changed part: $\tanh\left(x\right)^{1} \;\Longrightarrow\; \tanh\left(x\right)$
🧮 Current expression
$$2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\tanh\left(x\right)^{1}\,}}}\,\frac{d}{dx}{\left[\tanh\left(x\right)\right]}$$
✨ After: Power with exponent one
$$2\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\tanh\left(x\right)\,}}}\,\frac{d}{dx}{\left[\tanh\left(x\right)\right]}$$
Step 4 — 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.

Changed part: $\frac{d}{dx}{\left[\tanh\left(x\right)\right]} \;\Longrightarrow\; \frac{1}{\cosh\left(x\right)^{2}}$
How does this rule work? Wikipedia
🧮 Current expression
$$2\,\tanh\left(x\right)\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\tanh\left(x\right)\right]}\,}}}$$
✨ After: Derivative of the hyperbolic tangent
$$2\,\tanh\left(x\right) \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\cosh\left(x\right)^{2}}\,}}}$$
Step 5 — Combine into one fraction Simplification
📖 Rule
$$a\cdot\frac{b}{c}=\frac{a\cdot b}{c}$$

Why: A factor in front of a fraction multiplies its numerator, giving a single fraction.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\tanh\left(x\right) \cdot \frac{1}{\cosh\left(x\right)^{2}}\,}}}$$
✨ After: Combine into one fraction
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{2\,\tanh\left(x\right)}{\cosh\left(x\right)^{2}}\,}}}$$
Final result
$$\frac{2\,\tanh\left(x\right)}{\cosh\left(x\right)^{2}}$$
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Cross-check with a computer algebra system

The derivative computed directly by the computer algebra system Maxima:

$$2\,\operatorname{sech}\left(x\right)^{2}\,\tanh\left(x\right)$$
Step-by-step result ↔ Maxima: identical ✓