Differentiation Rules: Power Rule and Constant/Linearity Rules

Why rules matter (and what we are differentiating)

In this chapter you will build a toolkit for differentiating common algebraic expressions quickly and accurately. We will start with polynomials because they are made from powers of x and constants, and the rules are clean and reliable.

We will use several equivalent notations for the derivative:

• Function notation: if y = f(x), then the derivative is f'(x).
• Leibniz notation: if y depends on x, then the derivative is dy/dx.
• Operator view: d/dx means “differentiate with respect to x.” For example, d/dx (x^3).

All of these represent the same idea: a new function that gives the instantaneous rate of change of the original function.

Core linearity rules

1) Constant rule

If c is a constant (a number), then

d/dx (c) = 0

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Examples:

• d/dx (7) = 0
• If f(x)= -3, then f'(x)=0

2) Constant multiple rule

If c is a constant and g(x) is differentiable, then

d/dx (c·g(x)) = c·g'(x)

Example (step-by-step):

• Differentiate y = 5x^4
• Think: y = 5·(x^4)
• So dy/dx = 5·d/dx(x^4)

3) Sum and difference rule

If f and g are differentiable, then

d/dx (f(x) + g(x)) = f'(x) + g'(x)

d/dx (f(x) - g(x)) = f'(x) - g'(x)

This is why polynomials are friendly: you can differentiate term-by-term.

The power rule (the workhorse)

Power rule statement

For any real number n,

d/dx (x^n) = n·x^(n-1)

In function notation: if f(x)=x^n, then f'(x)=n x^(n-1).

What to check every time (quick self-audit)

• Exponent decreases by 1: n → n-1
• Coefficient multiplies by the old exponent: new coefficient is n times the old coefficient
• Constants vanish: derivative of a standalone number is 0

Scaffold 1: Differentiate monomials

A monomial looks like a x^n, where a is a constant and n is a real number.

Template

If y = a x^n, then

dy/dx = a·n·x^(n-1)

Examples (step-by-step)

Example 1: Differentiate y = x^5

• Old exponent: 5
• Multiply coefficient by 5 (coefficient is 1): 5
• Decrease exponent by 1: x^4

dy/dx = 5x^4

Example 2: Differentiate f(x) = -3x^2

• Old coefficient: -3, old exponent: 2
• New coefficient: -3·2 = -6
• New exponent: 2-1 = 1

f'(x) = -6x

Example 3: Differentiate y = 12x

• Rewrite as 12x^1
• Apply power rule: dy/dx = 12·1·x^0
• Use x^0 = 1

dy/dx = 12

Example 4: Differentiate y = 9x^0

• Note 9x^0 = 9 (a constant)
• Derivative is 0

dy/dx = 0

Practice: monomials (with immediate feedback guidance)

• 1. d/dx (x^8)
  Check: coefficient becomes 8; exponent becomes 7.
• 2. d/dx (-5x^3)
  Check: multiply -5 by 3; exponent drops to 2.
• 3. d/dx (4x)
  Check: treat as 4x^1; result should be a constant.
• 4. d/dx (11)
  Check: constants differentiate to 0.

Scaffold 2: Differentiate polynomials term-by-term

A polynomial is a sum of monomials with nonnegative integer exponents, such as 3x^4 - 2x + 7. Use linearity (sum/difference and constant multiple rules) to differentiate each term separately.

Example (fully worked)

Differentiate f(x) = 3x^4 - 2x + 7.

• Differentiate 3x^4: 3·4x^3 = 12x^3
• Differentiate -2x: rewrite as -2x^1 so derivative is -2·1·x^0 = -2
• Differentiate 7: 0

So f'(x) = 12x^3 - 2.

Example (watch the signs)

Differentiate y = -x^5 + 6x^2 - 4x + 9.

• d/dx(-x^5) = -5x^4
• d/dx(6x^2) = 12x
• d/dx(-4x) = -4
• d/dx(9) = 0

dy/dx = -5x^4 + 12x - 4

Practice: polynomials (with immediate feedback guidance)

• 1. d/dx (2x^6 + x^3 - 10)
  Check: each term separately; constant becomes 0.
• 2. d/dx (7x^4 - 3x^2 + x)
  Check: the x term becomes 1 (since it is x^1).
• 3. d/dx (-4x^3 - 2x + 5)
  Check: keep the negative sign attached to the coefficient.

Rewrite before differentiating (to reduce mistakes)

Many errors come from trying to apply the power rule to expressions that are not written as powers. A reliable habit is: rewrite using exponents first, then differentiate.

Common rewrites

• 1/x^3 = x^(-3)
• 1/√x = x^(-1/2)
• √x = x^(1/2)
• ∛(x^2) = x^(2/3)

Example: rewrite a reciprocal power

Differentiate y = 1/x^3.

• Rewrite: y = x^(-3)
• Apply power rule: dy/dx = -3·x^(-4)
• Optional rewrite back: -3/x^4

dy/dx = -3x^(-4) = -3/x^4

Example: rewrite a radical

Differentiate f(x) = √x.

• Rewrite: f(x) = x^(1/2)
• Power rule: f'(x) = (1/2)x^(-1/2)
• Optional rewrite: f'(x) = 1/(2√x)

Scaffold 3: Negative and fractional exponents

The power rule still works when the exponent is negative or fractional. The same two checks apply: multiply by the old exponent, then subtract 1 from the exponent.

Negative exponent examples

Example 1: Differentiate y = x^(-2)

• Multiply by old exponent: -2
• Decrease exponent by 1: -2 - 1 = -3

dy/dx = -2x^(-3) = -2/x^3

Example 2: Differentiate y = 5x^(-1)

• Constant multiple rule: keep 5
• Power rule: derivative of x^(-1) is -1·x^(-2)

dy/dx = -5x^(-2) = -5/x^2

Fractional exponent examples

Example 1: Differentiate y = x^(3/2)

• Multiply by old exponent: 3/2
• Decrease exponent by 1: 3/2 - 1 = 1/2

dy/dx = (3/2)x^(1/2)

Example 2: Differentiate y = x^(-1/2)

• Multiply by old exponent: -1/2
• Decrease exponent by 1: -1/2 - 1 = -3/2

dy/dx = (-1/2)x^(-3/2)

Practice: negative and fractional exponents (with immediate feedback guidance)

• 1. d/dx (x^(-5))
  Check: coefficient becomes -5; exponent becomes -6.
• 2. d/dx (3x^(-2))
  Check: new coefficient is 3·(-2); exponent becomes -3.
• 3. d/dx (x^(1/3))
  Check: exponent becomes -2/3 after subtracting 1.
• 4. d/dx (2/√x)
  Check: rewrite as 2x^(-1/2) first, then apply the power rule.

Putting the rules together: a mixed example

Differentiate f(x) = 4x^3 - 2/x^2 + 7√x - 9.

Step 1: Rewrite using exponents.

• -2/x^2 = -2x^(-2)
• 7√x = 7x^(1/2)

So f(x) = 4x^3 - 2x^(-2) + 7x^(1/2) - 9.

Step 2: Differentiate term-by-term.

• d/dx(4x^3) = 12x^2
• d/dx(-2x^(-2)) = -2·(-2)x^(-3) = 4x^(-3)
• d/dx(7x^(1/2)) = 7·(1/2)x^(-1/2) = (7/2)x^(-1/2)
• d/dx(-9) = 0

f'(x) = 12x^2 + 4x^(-3) + (7/2)x^(-1/2)

Optional rewrite: f'(x) = 12x^2 + 4/x^3 + 7/(2√x)

Common error checklist (use while practicing)

More practice (mixed, short)

• 1. If f(x)=x^4-3x^2+2, find f'(x).
  Check: exponents 4→3 and 2→1; constant disappears.
• 2. Find d/dx (6x^(-3) + x^(1/2)).
  Check: new coefficients: 6·(-3) and 1/2; exponents: -3→-4 and 1/2→-1/2.
• 3. Differentiate y = (5/x) - 2√x.
  Check: rewrite as 5x^(-1) - 2x^(1/2) first.
• 4. Differentiate y = -x + 8.
  Check: derivative should be a constant; the 8 should vanish.