Derivative Calculator

Derivative of $\arctan\left(\frac{x}{3}\right)$

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

Step-by-step solution

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

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

Step 1 — Derivative of Arctangent Differentiation rule
📖 Rule
$$\frac{d}{dx}\arctan\left(u\right)=\frac{1}{1 + u^{2}}\cdot u'$$
With:
  • $u = \frac{x}{3}$

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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\arctan\left(\frac{x}{3}\right)\right]}\,}}}$$
✨ After: Derivative of Arctangent
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{1 + \left(\frac{x}{3}\right)^{2}}\,\frac{d}{dx}{\left[\frac{x}{3}\right]}\,}}}$$
Step 2 — Divide by a constant Rewriting
📖 Rule
$$\frac{u}{c}=\frac{1}{c}\cdot u\quad(c\ne0)$$

Why: Dividing by a constant means multiplying by its reciprocal. First write that factor explicitly. Apply the constant factor rule in the next step.

Changed part: $\frac{d}{dx}{\left[\frac{x}{3}\right]} \;\Longrightarrow\; \frac{d}{dx}{\left[\frac{1}{3}\,x\right]}$
🧮 Current expression
$$\frac{1}{1 + \left(\frac{x}{3}\right)^{2}}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\frac{x}{3}\right]}\,}}}$$
✨ After: Divide by a constant
$$\frac{1}{1 + \left(\frac{x}{3}\right)^{2}}\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\frac{1}{3}\,x\right]}\,}}}$$
Step 3 — Constant Factor Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}(c\,u)=c\,u'$$
With:
  • $c = \frac{1}{3}$
  • $u = x$

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

Changed part: $\frac{d}{dx}{\left[\frac{1}{3}\,x\right]} \;\Longrightarrow\; \frac{1}{3}\,\frac{d}{dx}{\left[x\right]}$
How does this rule work? Wikipedia
🧮 Current expression
$$\frac{1}{1 + \left(\frac{x}{3}\right)^{2}}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\frac{1}{3}\,x\right]}\,}}}$$
✨ After: Constant Factor Rule
$$\frac{1}{1 + \left(\frac{x}{3}\right)^{2}}\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{3}\,\frac{d}{dx}{\left[x\right]}\,}}}$$
Step 4 — 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
$$\frac{1}{1 + \left(\frac{x}{3}\right)^{2}}\,\frac{1}{3}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\right]}\,}}}$$
✨ After: Derivative of a Variable
$$\frac{1}{1 + \left(\frac{x}{3}\right)^{2}}\,\frac{1}{3} \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}$$
Step 5 — 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: $\frac{1}{3} \cdot 1 \;\Longrightarrow\; \frac{1}{3}$
🧮 Current expression
$$\frac{1}{1 + \left(\frac{x}{3}\right)^{2}}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{3} \cdot 1\,}}}$$
✨ After: Remove a factor of one
$$\frac{1}{1 + \left(\frac{x}{3}\right)^{2}} \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{3}\,}}}$$
Step 6 — 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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{1 + \left(\frac{x}{3}\right)^{2}} \cdot \frac{1}{3}\,}}}$$
✨ After: Combine into one fraction
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\left(1 + \left(\frac{x}{3}\right)^{2}\right) \cdot 3}\,}}}$$
Step 7 — 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.

Changed part: $\left(1 + \left(\frac{x}{3}\right)^{2}\right) \cdot 3 \;\Longrightarrow\; 3\,\left(1 + \left(\frac{x}{3}\right)^{2}\right)$
🧮 Current expression
$$\frac{1}{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\left(1 + \left(\frac{x}{3}\right)^{2}\right) \cdot 3\,}}}}$$
✨ After: Order the factors
$$\frac{1}{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,3\,\left(1 + \left(\frac{x}{3}\right)^{2}\right)\,}}}}$$
Final result
$$\frac{1}{3\,\left(1 + \left(\frac{x}{3}\right)^{2}\right)}$$
Copy link to this solution
Cross-check with a computer algebra system

The derivative computed directly by the computer algebra system Maxima:

$$\frac{1}{3\,\left(\frac{x^{2}}{9} + 1\right)}$$
Step-by-step result ↔ Maxima: identical ✓