Differentiation Rules Formula

The Formula

Power: \frac{d}{dx}[x^n] = nx^{n-1}. Product: (fg)' = f'g + fg'. Quotient: \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}.

When to use: Shortcuts so you don't have to use the limit definition every time.

Quick Example

Power rule: \frac{d}{dx}(x^n) = nx^{n-1} So \frac{d}{dx}(x^3) = 3x^2.

Notation

(fg)' for product rule, \left(\frac{f}{g}\right)' for quotient rule. Prime notation f' or Leibniz notation \frac{d}{dx}[f].

What This Formula Means

A set of standard formulas for finding derivatives of common function types without using the limit definition each time.

Shortcuts so you don't have to use the limit definition every time.

Formal View

Power: \frac{d}{dx}[x^n] = nx^{n-1} for n \in \mathbb{R}. Product: (fg)'(x) = f'(x)g(x) + f(x)g'(x). Quotient: \left(\frac{f}{g}\right)'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2},\; g(x) \neq 0.

Worked Examples

Example 1

easy
Use the product rule to differentiate f(x) = x^3 \cdot \sin x.

Solution

  1. 1
    Identify the two factors: u = x^3 and v = \sin x.
  2. 2
    Find their derivatives: u' = 3x^2 and v' = \cos x.
  3. 3
    Apply the product rule (uv)' = u'v + uv'.
  4. 4
    Result: f'(x) = 3x^2 \sin x + x^3 \cos x.

Answer

f'(x) = 3x^2 \sin x + x^3 \cos x
The product rule states (fg)' = f'g + fg'. Each factor is differentiated once while the other is kept intact, and the two results are added. Never multiply the individual derivatives together.

Example 2

medium
Use the quotient rule to differentiate f(x) = \dfrac{x^2 + 1}{x - 3}.

Common Mistakes

  • Applying the quotient rule with the terms in the wrong order: it's \frac{f'g - fg'}{g^2} (lo-d-hi minus hi-d-lo), not the other way around.
  • Forgetting to apply the chain rule when using the power rule on expressions like (3x+1)^5 — the derivative is 5(3x+1)^4 \cdot 3, not 5(3x+1)^4.
  • Treating \frac{d}{dx}[e^x] as xe^{x-1} by misapplying the power rule — e^x is not a power function, its derivative is e^x.

Why This Formula Matters

Differentiation rules make finding derivatives fast and practical for polynomials, products, quotients, and composites.

Frequently Asked Questions

What is the Differentiation Rules formula?

A set of standard formulas for finding derivatives of common function types without using the limit definition each time.

How do you use the Differentiation Rules formula?

Shortcuts so you don't have to use the limit definition every time.

What do the symbols mean in the Differentiation Rules formula?

(fg)' for product rule, \left(\frac{f}{g}\right)' for quotient rule. Prime notation f' or Leibniz notation \frac{d}{dx}[f].

Why is the Differentiation Rules formula important in Math?

Differentiation rules make finding derivatives fast and practical for polynomials, products, quotients, and composites.

What do students get wrong about Differentiation Rules?

Product rule: (fg)' = f'g + fg', NOT (fg)' = f'g' — you cannot just multiply the individual derivatives.

What should I learn before the Differentiation Rules formula?

Before studying the Differentiation Rules formula, you should understand: derivative.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Derivatives Explained: Rules, Interpretation, and Applications →
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