Derivative of 1/x

Q: Is the derivative of 1/x equal to ln(x)?

No – it is the other way round: the derivative of \ln x is \frac1x. The derivative of \frac1x is -\frac{1}{x^2}.

The derivative of the reciprocal function is $\dfrac{d}{dx}\dfrac{1}{x} = -\dfrac{1}{x^2}$ for $x \neq 0$.

$$ \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2} $$

See the derivative of 1/x step by step in the calculator →

📈 Graph of 1/x and its derivative

The hyperbola $\frac1x$ falls on both branches, so its derivative is negative everywhere. Close to $x=0$ the curve is very steep ($-\frac{1}{x^2}$ is large), far away it is almost flat.

🔍 Why is the derivative of 1/x equal to $-\frac{1}{x^2}$?

Power rule: $\frac1x = x^{-1}$, so the derivative is $-1\cdot x^{-2} = -\frac{1}{x^2}$.

Quotient rule: $u = 1$, $v = x$: $\;\frac{0\cdot x - 1\cdot 1}{x^2} = -\frac{1}{x^2}$.

Definition of the derivative:

$$ \lim_{h\to0}\frac{\frac{1}{x+h}-\frac1x}{h} = \lim_{h\to0}\frac{x-(x+h)}{h\,x(x+h)} = \lim_{h\to0}\frac{-1}{x(x+h)} = -\frac{1}{x^2}. $$

🔗 Derivative of 1/x with an inner function

If the argument is a function $u(x)$, the chain rule adds the inner derivative:

$$ \frac{d}{dx}\frac{1}{u(x)} = -\frac{u'(x)}{u(x)^2} $$

✏️ Worked examples

Function	Derivative	Note
$\frac{1}{x^2}$	$-\frac{2}{x^3}$	$x^{-2}$ → $-2x^{-3}$.	steps →
$\frac{3}{x}$	$-\frac{3}{x^2}$	Constant factor rule.	steps →
$\frac{1}{2x+1}$	$-\frac{2}{(2x+1)^2}$	Reciprocal rule with $u = 2x+1$.	steps →
$\frac{1}{x} - x$	$-\frac{1}{x^2} - 1$	Sum rule.	steps →

⚠️ Common mistakes

Writing $\ln x$ – that is the antiderivative of $\frac1x$, not its derivative.

Getting $-\frac{1}{x}$ by lowering the exponent the wrong way: $-1-1 = -2$.

❓ Frequently asked questions

What is the derivative of 1/x²?

$-\frac{2}{x^3}$.

Is the derivative of 1/x equal to ln(x)?

No – it is the other way round: the derivative of $\ln x$ is $\frac1x$. The derivative of $\frac1x$ is $-\frac{1}{x^2}$.

What is the second derivative of 1/x?

$\frac{2}{x^3}$.