Derivative Rules

Learn power, product, quotient, and chain rules as reusable rate patterns.

The derivative definition explains what a derivative means, but using the limit definition for every problem would be slow. Derivative rules are reliable shortcuts built from patterns that appear again and again.

The hardest part for many students is not the arithmetic. It is deciding which rule to use first.

Learning objectives

After this lesson you should be able to:

  • apply constant, sum, difference, and constant-multiple rules;
  • use the power rule with integer, negative, and fractional powers;
  • recognize product, quotient, and chain rule situations;
  • combine several derivative rules in one expression;
  • diagnose common derivative-rule mistakes;
  • check derivatives with SymPy and edumath validators.

Rule selection roadmap

Before differentiating, ask these questions in order.

  1. Is the expression a constant? Use the constant rule.
  2. Is it a sum or difference? Differentiate term by term.
  3. Is there a constant multiplier? Keep the constant outside.
  4. Is it a simple power of \(x\)? Use the power rule.
  5. Are two changing factors multiplied? Use the product rule.
  6. Is one changing expression divided by another? Use the quotient rule.
  7. Is there an outside function applied to an inside function? Use the chain rule.
  8. Are several patterns present? Start with the outermost operation.

Rule choice comes before calculation

A good derivative solution begins with a reason: “I use the chain rule because this is an outside function applied to an inside function,” or “I use the product rule because two changing factors are multiplied.”

Linearity rules

Constants

The derivative of a constant is zero:

\[ \frac{d}{dx}[7]=0. \]

A constant function has no change, so its rate of change is zero.

Constant multiples

Constants multiply the derivative:

\[ \frac{d}{dx}[5f(x)]=5f'(x). \]

Sums and differences

Differentiate term by term:

\[ \frac{d}{dx}[f(x)+g(x)]=f'(x)+g'(x). \]

Walkthrough: term-by-term derivative

Differentiate

\[ 4x^3-2x+9. \]

  1. Classify. This is a sum/difference of terms.
  2. Differentiate each term.
    • \(4x^3\) becomes \(12x^2\).
    • \(-2x\) becomes \(-2\).
    • \(9\) becomes \(0\).
  3. Combine. \[ \frac{d}{dx}(4x^3-2x+9)=12x^2-2. \]
  4. Check. The constant disappeared because constant functions have zero slope.

Power rule

For many powers,

\[ \frac{d}{dx}x^n=nx^{n-1}. \]

The exponent moves down as a multiplier, then decreases by 1.

Examples:

\[ \frac{d}{dx}x^6=6x^5, \qquad \frac{d}{dx}x^{-1}=-x^{-2}, \qquad \frac{d}{dx}x^{1/2}=\frac12x^{-1/2}. \]

Guided exercise: power rule forms

Differentiate each expression.

  1. \(x^5\)
  2. \(7x^4\)
  3. \(1/x\), rewritten as \(x^{-1}\)
  4. \(\sqrt{x}\), rewritten as \(x^{1/2}\)
  1. \(5x^4\).
  2. \(28x^3\) because the constant 7 stays outside.
  3. \(-x^{-2}=-1/x^2\).
  4. \(\frac12x^{-1/2}=\frac{1}{2\sqrt{x}}\).

Product rule

Use the product rule when two changing factors are multiplied:

\[ \frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x). \]

Walkthrough: product rule

Differentiate

\[ (x^2+1)(x-3). \]

  1. Identify factors. \[ f(x)=x^2+1, \qquad g(x)=x-3. \]
  2. Differentiate each factor. \[ f'(x)=2x, \qquad g'(x)=1. \]
  3. Substitute into product rule. \[ f'g+fg'=2x(x-3)+(x^2+1)(1). \]
  4. Simplify if useful. \[ 2x^2-6x+x^2+1=3x^2-6x+1. \]
  5. Check structure. Both factors were differentiated once; no factor was forgotten.

Quotient rule

Use the quotient rule when one changing expression is divided by another:

\[ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] =\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}. \]

A memory phrase some students use is: “low times derivative of high minus high times derivative of low, over low squared.”

Walkthrough: quotient rule

Differentiate

\[ \frac{x^2+1}{x-1}. \]

  1. Identify numerator and denominator. \[ f(x)=x^2+1, \qquad g(x)=x-1. \]
  2. Differentiate both. \[ f'(x)=2x, \qquad g'(x)=1. \]
  3. Substitute carefully. \[ \frac{2x(x-1)-(x^2+1)(1)}{(x-1)^2}. \]
  4. Simplify numerator. \[ 2x^2-2x-x^2-1=x^2-2x-1. \]
  5. Final derivative. \[ \frac{x^2-2x-1}{(x-1)^2}. \]
WarningQuotient-rule mistake

The order in the numerator matters. Reversing the subtraction changes the sign of the answer.

Chain rule

Use the chain rule for a composition: an outside function applied to an inside function.

\[ \frac{d}{dx}F(g(x))=F'(g(x))g'(x). \]

Walkthrough: chain rule

Differentiate

\[ (x^2+1)^3. \]

  1. Identify the outside function. The outside is “cube something.”
  2. Identify the inside function. The inside is \(x^2+1\).
  3. Differentiate the outside, leaving the inside in place. \[ 3(x^2+1)^2. \]
  4. Multiply by the derivative of the inside. \[ \frac{d}{dx}(x^2+1)=2x. \]
  5. Combine. \[ 3(x^2+1)^2(2x)=6x(x^2+1)^2. \]

Another chain rule example

\[ \frac{d}{dx}\sin(x^2)=\cos(x^2)\cdot 2x. \]

The factor \(2x\) is the derivative of the inside. Forgetting this factor is the most common chain-rule error.

Combining rules

For

\[ h(x)=(x^2+1)^3e^x, \]

the outermost operation is multiplication of two changing factors. So use product rule first. The first factor then needs chain rule.

  1. Let \(f(x)=(x^2+1)^3\) and \(g(x)=e^x\).
  2. \(f'(x)=6x(x^2+1)^2\) by chain rule.
  3. \(g'(x)=e^x\).
  4. Product rule gives \[ h'(x)=6x(x^2+1)^2e^x+(x^2+1)^3e^x. \]

Mistake diagnosis

A student differentiates \((x^2+1)^3\) and writes \(3(x^2+1)^2\).

  1. Which rule were they using?
  2. What did they forget?
  3. What is the corrected derivative?
  1. They were using the chain rule.
  2. They forgot the derivative of the inside function, \(2x\).
  3. The corrected derivative is \(3(x^2+1)^2(2x)=6x(x^2+1)^2\).

Mixed practice

  1. Differentiate \(x^6\).
  2. Differentiate \(4x^3-2x+9\).
  3. Which rule comes first for \((x^2+1)^5\)?
  4. Which rule comes first for \((x^2+1)(x-4)\)?
  5. Differentiate \((x^2+1)^3\).
  6. Differentiate \(1/x\) by rewriting it as a power.
  7. Differentiate \(x e^x\).
  8. Differentiate \(\sin(x^2)\).
  1. \(6x^5\).
  2. \(12x^2-2\).
  3. Chain rule.
  4. Product rule.
  5. \(6x(x^2+1)^2\).
  6. \(1/x=x^{-1}\), so the derivative is \(-x^{-2}=-1/x^2\).
  7. Product rule: \(1\cdot e^x+x e^x=e^x(1+x)\).
  8. \(2x\cos(x^2)\).

Guessing game checkpoint

Using this lesson with edumath and SymPy

SymPy can check equivalent derivative forms. This is useful because correct answers may look different after expansion or factoring.

from edumath.calculus import derivative, derivative_rule_trace, validate_derivative_equivalence

expected = derivative("(x**2 + 1)**3")
expected

\(\displaystyle 6 x \left(x^{2} + 1\right)^{2}\)

validate_derivative_equivalence("6*x*(x**2 + 1)**2", expected).message
'Correct.'
derivative_rule_trace("(x**2 + 1)**3")
('chain rule: differentiate the outside power and multiply by the inside derivative',
 'sum/difference rule: differentiate each term',
 'constant rule: the expression does not depend on the variable',
 'power rule: bring down 2 and subtract 1')
import sympy as sp

x = sp.symbols("x")
sp.simplify(sp.diff((x**2 + 1) ** 3, x))

\(\displaystyle 6 x \left(x^{2} + 1\right)^{2}\)

Further reading