Derivative of 1/x- Calculus Rules and Solutions

What Is the Derivative of 1/x?

The derivative of 1/x is -1/x². That's it. No tricks, no hidden steps. If you need the derivative, that's your answer.

But if you want to understand why it's -1/x², or how to derive it using different calculus rules, keep reading. This guide covers every method, common mistakes, and practical examples.

The Basic Answer

For the function f(x) = 1/x, the derivative is:

f'(x) = -1/x²

This applies for all x ≠ 0. If x = 0, the function is undefined, so the derivative doesn't exist there either.

How to Find the Derivative of 1/x

There are three main ways to arrive at this answer. I'll show you each one.

Method 1: Power Rule (Fastest)

Rewrite 1/x as x⁻¹. Then apply the power rule:

d/dx [xⁿ] = n · xⁿ⁻¹

So:

d/dx [x⁻¹] = -1 · x⁻² = -1/x²

This is the fastest method. If you see 1/x anywhere, just convert it to x⁻¹ and apply the power rule.

Method 2: Quotient Rule

The quotient rule states:

d/dx [f(x)/g(x)] = [f'(x) · g(x) - f(x) · g'(x)] / [g(x)]²

For 1/x, treat it as 1/x¹:

• f(x) = 1, so f'(x) = 0
• g(x) = x, so g'(x) = 1

Plug in:

[0 · x - 1 · 1] / x² = -1/x²

Same answer. The quotient rule works, but it's unnecessary work for something this simple.

Method 3: Implicit Differentiation

Start with y = 1/x. Multiply both sides by x:

xy = 1

Now differentiate implicitly:

x · dy/dx + y · 1 = 0

Solve for dy/dx:

dy/dx = -y/x

Substitute back y = 1/x:

dy/dx = -(1/x)/x = -1/x²

This method is useful when you're working with implicit relationships, but it's overkill here.

Comparing the Methods

Use the power rule every time. It's the most direct path to the answer.

Examples and Solutions

Example 1: Basic Derivative

Find d/dx [1/x] at x = 3

d/dx [1/x] = -1/x²

At x = 3: -1/3² = -1/9

The slope of the tangent line to y = 1/x at x = 3 is -1/9.

Example 2: Chain Rule with Composite Function

Find d/dx [1/(3x + 1)]

Let u = 3x + 1. Then 1/(3x + 1) = u⁻¹

Apply chain rule:

d/dx [u⁻¹] = -1 · u⁻² · du/dx

d/dx = -1/(3x + 1)² · 3

d/dx = -3/(3x + 1)²

Example 3: Product Rule

Find d/dx [x² · (1/x)]

Simplify first: x² · (1/x) = x

The derivative is simply 1.

If you didn't simplify, use the product rule:

d/dx [x²] · (1/x) + x² · d/dx [1/x]

= 2x · (1/x) + x² · (-1/x²)

= 2 - 1 = 1

Same answer. Simplification saves time.

Common Mistakes to Avoid

• Forgetting the negative sign. The derivative of x⁻¹ is always -x⁻². No exceptions.
• Treating 1/x as x without a reciprocal. d/dx[x] = 1, not -1/x².
• Domain errors. Remember: x ≠ 0 for both the function and its derivative.
• Overcomplicating with the quotient rule. When a function is already a simple power, use the power rule.

Higher Order Derivatives

The second derivative of 1/x:

f'(x) = -x⁻²

f''(x) = 2x⁻³ = 2/x³

The third derivative:

f'''(x) = -6x⁻⁴ = -6/x⁴

Pattern: f⁽ⁿ⁾(x) = (-1)ⁿ · n! / xⁿ⁺¹

How to Get Started: Quick Practice

Try these problems to confirm you understand the concept:

1. Find the derivative of 1/x at x = -2
2. Find the derivative of 1/(x³)
3. Find d/dx [5/x]

Solutions:

1. f'(-2) = -1/(-2)² = -1/4
2. Rewrite as x⁻³: derivative is -3x⁻⁴ = -3/x⁴
3. Constant multiple rule: 5 · (-1/x²) = -5/x²

When You'll Use This

The derivative of 1/x shows up in:

• Reciprocal functions in physics (inverse relationships)
• Optimization problems involving inverse proportionality
• Rate of change problems where quantities vary inversely
• Integration problems that require the 1/x antiderivative (ln|x|)

That last point matters. If you're working through integration later, knowing that ∫(1/x) dx = ln|x| + C connects directly to this derivative.

The Bottom Line

The derivative of 1/x is -1/x². Use the power rule. Rewrite 1/x as x⁻¹, apply d/dx[xⁿ] = n·xⁿ⁻¹, and you're done.

Don't overthink it. Don't use the quotient rule unless you have to. The power rule exists for exactly this situation.