Derivative Calculator

Derivative of $\frac{1}{2\,x + 1}$

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

Step-by-step solution

Input function
$$\frac{1}{2\,x + 1}$$
Start of the differentiation
$$\frac{d}{dx}{\left[\frac{1}{2\,x + 1}\right]}$$

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

Step 1 — Write the reciprocal as a negative power Rewriting
📖 Rule
$$\frac{c}{v}=c\cdot v^{-1},\quad \frac{c}{v^k}=c\cdot v^{-k}$$

Why: Write the denominator as a power with a negative exponent, then apply the factor and power rules. The original denominator must still be nonzero.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\frac{1}{2\,x + 1}\right]}\,}}}$$
✨ After: Write the reciprocal as a negative power
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\left(2\,x + 1\right)^{-1}\right]}\,}}}$$
Step 2 — Power Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}\,u'$$
With:
  • $u = 2\,x + 1$
  • $c = -1$

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

Why: Evaluate only the highlighted numeric expression, preserving exact fractions and decimals.

Changed part: $-1 - 1 \;\Longrightarrow\; -2$
🧮 Current expression
$$-1\,\left(2\,x + 1\right)^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-1 - 1\,}}}}\,\frac{d}{dx}{\left[2\,x + 1\right]}$$
✨ After: Evaluate numbers
$$-1\,\left(2\,x + 1\right)^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-2\,}}}}\,\frac{d}{dx}{\left[2\,x + 1\right]}$$
Step 4 — Sum/Difference Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}(u \pm v \pm \dots) = \frac{d}{dx}(u) \pm \frac{d}{dx}(v) \pm \dots$$
With:
  • $u = 2\,x$
  • $v = 1$

Why: Differentiate every term separately and keep the plus and minus signs between them. Even a constant term first receives its own derivative symbol.

Changed part: $\frac{d}{dx}{\left[2\,x + 1\right]} \;\Longrightarrow\; \frac{d}{dx}{\left[2\,x\right]} + \frac{d}{dx}{\left[1\right]}$
How does this rule work? Wikipedia
🧮 Current expression
$$-1\,\left(2\,x + 1\right)^{-2}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[2\,x + 1\right]}\,}}}$$
✨ After: Sum/Difference Rule
$$-1\,\left(2\,x + 1\right)^{-2}\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\left(\frac{d}{dx}{\left[2\,x\right]} + \frac{d}{dx}{\left[1\right]}\right)\,}}}$$
Step 5 — Constant Factor Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}(c\,u)=c\,u'$$
With:
  • $c = 2$
  • $u = x$

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

Changed part: $\frac{d}{dx}{\left[2\,x\right]} \;\Longrightarrow\; 2\,\frac{d}{dx}{\left[x\right]}$
How does this rule work? Wikipedia
🧮 Current expression
$$-1\,\left(2\,x + 1\right)^{-2}\,\left(\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[2\,x\right]}\,}}} + \frac{d}{dx}{\left[1\right]}\right)$$
✨ After: Constant Factor Rule
$$-1\,\left(2\,x + 1\right)^{-2}\,\left(\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\frac{d}{dx}{\left[x\right]}\,}}} + \frac{d}{dx}{\left[1\right]}\right)$$
Step 6 — 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
$$-1\,\left(2\,x + 1\right)^{-2}\,\left(2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\right]}\,}}} + \frac{d}{dx}{\left[1\right]}\right)$$
✨ After: Derivative of a Variable
$$-1\,\left(2\,x + 1\right)^{-2}\,\left(2 \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}} + \frac{d}{dx}{\left[1\right]}\right)$$
Step 7 — 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: $2 \cdot 1 \;\Longrightarrow\; 2$
🧮 Current expression
$$-1\,\left(2\,x + 1\right)^{-2}\,\left(\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 \cdot 1\,}}} + \frac{d}{dx}{\left[1\right]}\right)$$
✨ After: Remove a factor of one
$$-1\,\left(2\,x + 1\right)^{-2}\,\left(\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,}}} + \frac{d}{dx}{\left[1\right]}\right)$$
Step 8 — Constant Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}(c)=0$$
With:
  • $c = 1$

Why: This term does not depend on x. Its value stays unchanged when x changes, so its derivative is zero. Other letters are treated as constant parameters.

Changed part: $\frac{d}{dx}{\left[1\right]} \;\Longrightarrow\; 0$
How does this rule work?
🧮 Current expression
$$-1\,\left(2\,x + 1\right)^{-2}\,\left(2 + \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[1\right]}\,}}}\right)$$
✨ After: Constant Rule
$$-1\,\left(2\,x + 1\right)^{-2}\,\left(2 + \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,0\,}}}\right)$$
Step 9 — Remove a zero term Simplification
📖 Rule
$$a+0=a,\quad a-0=a$$

Why: Adding or subtracting zero leaves the value unchanged. Remove that zero term.

Changed part: $2 + 0 \;\Longrightarrow\; 2$
🧮 Current expression
$$-1\,\left(2\,x + 1\right)^{-2}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\left(2 + 0\right)\,}}}$$
✨ After: Remove a zero term
$$-1\,\left(2\,x + 1\right)^{-2} \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,}}}$$
Step 10 — Write the factor −1 as a minus sign Simplification
📖 Rule
$$(-1)\cdot a=-a$$

Why: Multiplying by −1 gives the negative. Write just a minus sign instead.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-1\,\left(2\,x + 1\right)^{-2} \cdot 2\,}}}$$
✨ After: Write the factor −1 as a minus sign
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\left(2\,x + 1\right)^{-2} \cdot 2\,}}}$$
Step 11 — 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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\left(2\,x + 1\right)^{-2} \cdot 2\,}}}$$
✨ After: Order the factors
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-2\,\left(2\,x + 1\right)^{-2}\,}}}$$
Step 12 — Write negative exponents as a fraction Simplification
📖 Rule
$$a^{-n}=\frac{1}{a^n}$$

Why: A power with a negative exponent is the reciprocal of the power with the positive exponent. The result then has no negative exponents.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-2\,\left(2\,x + 1\right)^{-2}\,}}}$$
✨ After: Write negative exponents as a fraction
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{2}{\left(2\,x + 1\right)^{2}}\,}}}$$
Final result
$$-\frac{2}{\left(2\,x + 1\right)^{2}}$$
Valid where:
  • $2\,x + 1\ne0$ – The original denominator must be nonzero.

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:

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