Power Rule (Derivatives)

📖 What is the Power Rule?

The power rule is the most fundamental rule in differential calculus. It allows you to differentiate any term with a variable raised to a power. To apply it: multiply by the exponent, then reduce the exponent by one.

d/dx[xⁿ] = n·xⁿ⁻¹

🔤 What Each Part Means

d/dx

Derivative with respect to xThe rate of change of the function as x changes

xⁿ

The functionx raised to the power n

n

The exponentCan be any real number: positive, negative, or fraction

📝 Step-by-Step Examples

1

d/dx[x³]

Multiply by exponent, reduce by 1

= 3x²

2

d/dx[5x⁴]

Constant multiplies through

= 5 × 4x³ = 20x³

3

d/dx[x⁻²]

Works with negative exponents too

= -2x⁻³ = -2/x³

4

d/dx[√x] = d/dx[x^(1/2)]

Works with fractional exponents

= (1/2)x^(-1/2) = 1/(2√x)

💡 Special Cases

Constant rule:

d/dx[c] = 0    (constants have zero rate of change)

Linear rule:

d/dx[x] = 1    (slope of y = x is always 1)

Integration (reverse):

∫xⁿ dx = xⁿ⁺¹/(n+1) + C

Power Rule (Derivatives)

📖 What is the Power Rule?

🔤 What Each Part Means

📝 Step-by-Step Examples

💡 Special Cases

🔗 Related

🎯 Quick Fact