How to Calculate Derivatives - Power Rule Step by Step

Staring at f(x) = 3x² + 2x − 5 and wondering what f'(x) even means? You're not alone. Derivatives look intimidating, but the actual calculation is shockingly simple once you know the rules.

Here's how to find any derivative — no panic required.

What Is a Derivative, Really?

A derivative tells you the rate of change of a function at any point. Think of it as the slope of the tangent line — if you zoom in far enough on any curve, it looks like a straight line. The derivative gives you the slope of that line.

f'(x) = slope of f(x) at point x

If f(x) = x², then f'(x) = 2x. At x = 3, the slope is 6. At x = 5, the slope is 10. The curve gets steeper as x increases.

The Power Rule — Your #1 Weapon

The power rule handles 90% of the derivatives you'll see in school:

d/dx(xⁿ) = n · xⁿ⁻¹

Translation: bring the exponent down as a multiplier, then reduce the exponent by 1.

Step-by-Step Example

Find the derivative of f(x) = 4x³ + 7x² − 3x + 2

| Term | Apply Power Rule | Result | |------|-----------------|--------| | 4x³ | 4 · 3 · x² | 12x² | | 7x² | 7 · 2 · x¹ | 14x | | −3x | −3 · 1 · x⁰ | −3 | | +2 | constant → 0 | 0 |

f'(x) = 12x² + 14x − 3

That's it. Each term differentiates independently (sum rule), and the constant vanishes.

Evaluating the Derivative at a Point

The derivative itself is a function. But often you need the slope at a specific point.

Using f(x) = 4x³ + 7x² − 3x + 2, find the slope at x = 1:

1. f'(x) = 12x² + 14x − 3
2. f'(1) = 12(1)² + 14(1) − 3
3. f'(1) = 12 + 14 − 3 = 23

The slope of the tangent line at x = 1 is 23. The curve is rising steeply at that point.

Common Derivative Rules Cheat Sheet

| Rule | Formula | Example | |------|---------|---------| | Power Rule | d/dx(xⁿ) = nxⁿ⁻¹ | d/dx(x⁵) = 5x⁴ | | Constant Rule | d/dx(c) = 0 | d/dx(7) = 0 | | Coefficient Rule | d/dx(c·f) = c·f' | d/dx(3x²) = 6x | | Sum Rule | d/dx(f+g) = f'+g' | d/dx(x²+x) = 2x+1 | | Product Rule | d/dx(fg) = f'g + fg' | d/dx(x²·sinx) | | Chain Rule | d/dx(f(g(x))) = f'(g(x))·g'(x) | d/dx(sin(3x)) = 3cos(3x) |

The Trench Truth: Students lose more points on derivative exams from arithmetic errors than from misunderstanding the rules. Write out each term's derivative separately before combining them. Rushing leads to sign errors — that −3x becomes +3x and your entire answer is wrong.

Practice Problems

Try these yourself, then check with our derivative calculator:

1. f(x) = 5x⁴ − 2x³ + x → f'(x) = ?
2. f(x) = x⁷ + 3x² − 8 → f'(x) = ?
3. Find f'(2) when f(x) = 6x³ − x² + 4x

Related: Quadratic Formula Calculator · Integral Calculator · Square Root Calculator