Derivative Calculator

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

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

Step-by-step solution

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

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:
  • $u = \csc\left(x\right)$
  • $c = 2$

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

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

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

Changed part: $\frac{d}{dx}{\left[\csc\left(x\right)\right]} \;\Longrightarrow\; -\csc\left(x\right)\,\cot\left(x\right)$
How does this rule work? Wikipedia
🧮 Current expression
$$2\,\csc\left(x\right)\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\csc\left(x\right)\right]}\,}}}$$
✨ After: Derivative of the cosecant
$$2\,\csc\left(x\right)\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\left(-\csc\left(x\right)\,\cot\left(x\right)\right)\,}}}$$
Step 5 — Determine the sign of the product Simplification
📖 Rule
$$a\cdot(-b)=-(a\cdot b),\quad (-a)\cdot(-b)=a\cdot b$$

Why: Move the minus signs of the factors in front of the product. An even number of minus signs gives plus, an odd number gives minus.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\csc\left(x\right)\,\left(-\csc\left(x\right)\,\cot\left(x\right)\right)\,}}}$$
✨ After: Determine the sign of the product
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-2\,\csc\left(x\right)\,\csc\left(x\right)\,\cot\left(x\right)\,}}}$$
Step 6 — Write identical factors as a square Simplification
📖 Rule
$$a\cdot a=a^2$$

Why: An expression multiplied by itself is its square.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-2\,\csc\left(x\right)\,\csc\left(x\right)\,\cot\left(x\right)\,}}}$$
✨ After: Write identical factors as a square
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-2\,\csc\left(x\right)^{2}\,\cot\left(x\right)\,}}}$$
Final result
$$-2\,\csc\left(x\right)^{2}\,\cot\left(x\right)$$
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.

Copy link to this solution
Cross-check with a computer algebra system

The derivative computed directly by the computer algebra system Maxima:

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