The Power Rule (Derivatives)

It gives the slope of a power of x at any point: bring the exponent down in front, then lower the exponent by one.

ddx x^n = n x^ n-1

A derivative measures the slope of the curve at a point - the slope of the straight line that just touches it there.

What each symbol means

• x^n: the function being differentiated (x raised to a power n)
• n: the exponent (any fixed number)
• ddx: read "the derivative with respect to x": the instant rate of change, or slope

A derivative answers one question: how steep is a curve at a single point? Draw the straight line that just grazes the curve at that point (the tangent line). The derivative is that tangent line's slope. The power rule is the quickest way to find it when the function is a power of x.

The rule is short: to differentiate x^n, bring the exponent n down in front, then reduce the exponent by 1. That gives n x^n-1.

What "slope of a curve" means: A straight line has one slope everywhere. A curve is different: it is steeper in some places than others. The derivative gives you the slope at whichever point you choose.

A worked example

Differentiate x³

1. The exponent is n = 3.
2. Bring the exponent down in front: that gives a 3.
3. Lower the exponent by one: 3 - 1 = 2.
4. So ddx x³ = 3x².

A second example, with a coefficient

Differentiate 5x²

1. A constant multiplier just rides along, so keep the 5 and differentiate x².
2. Power rule on x²: bring down the 2, lower the exponent to 1, giving 2x^1 = 2x.
3. Multiply by the 5 that was waiting: 5 × 2x = 10x.
4. So ddx 5x² = 10x.

• A derivative is the slope of the curve at a point (the tangent line's slope).
• The power rule: ddx x^n = n x^n-1 (down in front, exponent minus one).
• A constant multiplier stays and rides along.
• The derivative of a plain constant is 0 (a flat line has no slope).

Where the power rule comes from (the idea)

1. The slope at a point is the rise over the run between that point and a second point very close by: f(x+h) - f(x)h, as the gap h shrinks toward 0.
2. For f(x) = x², work out f(x+h) = (x+h)² = x² + 2xh + h².
3. Subtract and divide by h: (x² + 2xh + h²) - x²h = 2x + h.
4. Now let h shrink to 0. The leftover is 2x, which is exactly n x^n-1 with n = 2. The same expansion for any power gives the general rule.

Practice questions

Question 1

Differentiate x^4.

1. Use the power rule: ddx x^n = n x^n-1.
2. The exponent is n = 4, so bring the 4 down in front.
3. Lower the exponent by one: 4 - 1 = 3.
4. So ddx x^4 = 4x³.

Answer: 4x³

Question 2

Differentiate x².

1. Use the power rule: ddx x^n = n x^n-1.
2. The exponent is n = 2, so bring the 2 down in front.
3. Lower the exponent by one: 2 - 1 = 1.
4. So ddx x² = 2x^1 = 2x.

Answer: 2x

Question 3

Differentiate 3x².

1. A constant multiplier stays put, so keep the 3 and differentiate x².
2. Power rule on x²: bring down the 2 and lower the exponent to 1, giving 2x.
3. Multiply by the 3 that was waiting: 3 × 2x = 6x.
4. So ddx 3x² = 6x.

Answer: 6x

Question 4

Differentiate x (that is x^1).

1. Rewrite x as x^1 so you can see the exponent.
2. Use the power rule: bring the 1 down in front.
3. Lower the exponent by one: 1 - 1 = 0, giving 1x^0.
4. Since x^0 = 1, this is 1x^0 = 1.

Answer: 1

Question 5

What is the derivative of the constant 7?

1. A constant like 7 graphs as a flat horizontal line.
2. A flat line never rises or falls, so its slope is the same everywhere.
3. The derivative is that slope, which is 0.
4. So the derivative of 7 is 0.

Answer: 0