Derivative of $\cot\left(x\right)$
Differentiated with respect to x. Edit the function · Differentiation rules
Step-by-step solution
Input function
$$\cot\left(x\right)$$
Start of the differentiation
$$\frac{d}{dx}{\left[\cot\left(x\right)\right]}$$
Yellow: the expression being replaced. Green: its replacement.
Step 1 — Derivative of the cotangent
Differentiation rule
📖 Rule
$$\frac{d}{dx}\cot\left(x\right)=-\frac{1}{\sin\left(x\right)^{2}}$$
Why: The argument is exactly x. Apply the known elementary derivative.
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\cot\left(x\right)\right]}\,}}}$$
✨ After: Derivative of the cotangent
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{1}{\sin\left(x\right)^{2}}\,}}}$$
Final result
$$-\frac{1}{\sin\left(x\right)^{2}}$$
Valid where:
- $\sin\left(x\right)\ne0$ – The function and its derivative are undefined at poles.
The derivative is only valid where the original function is defined.
Cross-check with a computer algebra system
The derivative computed directly by the computer algebra system Maxima:
$$-\csc\left(x\right)^{2}$$
Step-by-step result ↔ Maxima: identical ✓