Derivative Calculator

Derivative of $\csc\left(3\,x\right)$

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

Step-by-step solution

Input function
$$\csc\left(3\,x\right)$$
Start of the differentiation
$$\frac{d}{dx}{\left[\csc\left(3\,x\right)\right]}$$

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

Step 1 — Derivative of the cosecant (chain rule) Differentiation rule
📖 Rule
$$\frac{d}{dx}\csc\left(u\right)=-\csc\left(u\right)\,\cot\left(u\right)\cdot u'$$
With:
  • $u = 3\,x$

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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\csc\left(3\,x\right)\right]}\,}}}$$
✨ After: Derivative of the cosecant (chain rule)
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\csc\left(3\,x\right)\,\cot\left(3\,x\right)\,\frac{d}{dx}{\left[3\,x\right]}\,}}}$$
Step 2 — Constant Factor Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}(c\,u)=c\,u'$$
With:
  • $c = 3$
  • $u = x$

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

Changed part: $\frac{d}{dx}{\left[3\,x\right]} \;\Longrightarrow\; 3\,\frac{d}{dx}{\left[x\right]}$
How does this rule work? Wikipedia
🧮 Current expression
$$-\csc\left(3\,x\right)\,\cot\left(3\,x\right)\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[3\,x\right]}\,}}}$$
✨ After: Constant Factor Rule
$$-\csc\left(3\,x\right)\,\cot\left(3\,x\right) \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,3\,\frac{d}{dx}{\left[x\right]}\,}}}$$
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
$$-\csc\left(3\,x\right)\,\cot\left(3\,x\right) \cdot 3\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\right]}\,}}}$$
✨ After: Derivative of a Variable
$$-\csc\left(3\,x\right)\,\cot\left(3\,x\right) \cdot 3 \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}$$
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: $3 \cdot 1 \;\Longrightarrow\; 3$
🧮 Current expression
$$-\csc\left(3\,x\right)\,\cot\left(3\,x\right) \cdot \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,3 \cdot 1\,}}}$$
✨ After: Remove a factor of one
$$-\csc\left(3\,x\right)\,\cot\left(3\,x\right) \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,3\,}}}$$
Step 5 — Order the factors Simplification
📖 Rule
$$u\cdot c=c\cdot u$$

Why: The order of factors may be changed (commutative law). Usual order: number, constants, powers of the variable, other factors.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\csc\left(3\,x\right)\,\cot\left(3\,x\right) \cdot 3\,}}}$$
✨ After: Order the factors
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-3\,\csc\left(3\,x\right)\,\cot\left(3\,x\right)\,}}}$$
Final result
$$-3\,\csc\left(3\,x\right)\,\cot\left(3\,x\right)$$
Valid where:
  • $\sin\left(3\,x\right)\ne0$ – The function and its derivative are undefined at poles.

The derivative is only valid where the original function is defined.

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Cross-check with a computer algebra system

The derivative computed directly by the computer algebra system Maxima:

$$-3\,\cot\left(3\,x\right)\,\csc\left(3\,x\right)$$
Step-by-step result ↔ Maxima: identical ✓