Derivative Calculator

Derivative of $\tanh\left(2\,x\right)$

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

Step-by-step solution

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

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

Step 1 — Derivative of the hyperbolic tangent (chain rule) Differentiation rule
📖 Rule
$$\frac{d}{dx}\tanh\left(u\right)=\frac{1}{\cosh\left(u\right)^{2}}\cdot u'$$
With:
  • $u = 2\,x$

Why: Differentiate the outer function while keeping its inner argument u. Multiply by u' and evaluate that inner derivative in its own step.

How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\tanh\left(2\,x\right)\right]}\,}}}$$
✨ After: Derivative of the hyperbolic tangent (chain rule)
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\cosh\left(2\,x\right)^{2}}\,\frac{d}{dx}{\left[2\,x\right]}\,}}}$$
Step 2 — Constant Factor Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}(c\,u)=c\,u'$$
With:
  • $c = 2$
  • $u = x$

Why: Keep the constant multiplier outside the derivative. Next, differentiate only the part depending on x.

Changed part: $\frac{d}{dx}{\left[2\,x\right]} \;\Longrightarrow\; 2\,\frac{d}{dx}{\left[x\right]}$
How does this rule work? Wikipedia
🧮 Current expression
$$\frac{1}{\cosh\left(2\,x\right)^{2}}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[2\,x\right]}\,}}}$$
✨ After: Constant Factor Rule
$$\frac{1}{\cosh\left(2\,x\right)^{2}} \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\frac{d}{dx}{\left[x\right]}\,}}}$$
Step 3 — Derivative of a Variable Differentiation rule
📖 Rule
$$\frac{d}{dx}(x)=1$$

Why: When x increases by one, the value of this term also increases by one. Its derivative is therefore one.

Changed part: $\frac{d}{dx}{\left[x\right]} \;\Longrightarrow\; 1$
How does this rule work?
🧮 Current expression
$$\frac{1}{\cosh\left(2\,x\right)^{2}} \cdot 2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\right]}\,}}}$$
✨ After: Derivative of a Variable
$$\frac{1}{\cosh\left(2\,x\right)^{2}} \cdot 2 \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}$$
Step 4 — Remove a factor of one Simplification
📖 Rule
$$1\cdot a=a,\quad \frac{a}{1}=a$$

Why: Multiplication or division by one leaves the value unchanged.

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

The derivative computed directly by the computer algebra system Maxima:

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