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