Power, constant and sum rules

From limits to rules

Lesson 2.2.2 found derivatives from the limit: write the chord slope, simplify until $h$ cancels, then let $h \to 0$. That works for any function, but it is slow, and it gets harder with every new function. This lesson and the next three give rules that produce a derivative in one line.

$\frac{d}{dx}$ in front of an expression means "the derivative of this with respect to $x$". So $\frac{d}{dx}x^2 = 2x$ says the same as "if $f(x) = x^2$, then $f'(x) = 2x$". The units are output units per input unit, as in lesson 2.2.2.

The power rule

Lesson 2.2.2 worked out four derivatives from the limit and noticed a pattern. Written with every power showing:

Each time, the power comes down in front as a multiplier, and the power drops by one. The pattern holds for every power.

A check by numbers: $x^4$ at 2

The rule says the slope of $x^4$ is $4x^3$, so at $x = 2$ it is $4 \times 2^3 = 32$. The central difference from lesson 2.2.2 agrees.

Algebra explains the pattern. Expanding both powers, the central difference of $x^4$ at 2 is exactly $32 + 8h^2$, which tends to 32 as $h \to 0$. The limit of lesson 2.2.2 would need $(x + h)^4$ expanded; the rule needs one line.

Powers of every kind

• Whole powers, including 1 and 0. $x^1$ has slope $1 \cdot x^0 = 1$: the line $y = x$ has slope 1. $x^0 = 1$ has slope $0 \cdot x^{-1} = 0$: a flat line.
• Negative powers. Write $\frac{1}{x^2}$ as $x^{-2}$. Its derivative is $-2x^{-3} = -\frac{2}{x^3}$. The power goes from $-2$ down to $-3$, further from zero.
• Fractional powers. The cube root $\sqrt[3]{x} = x^{1/3}$ has derivative $\frac{1}{3}x^{-2/3} = \frac{1}{3x^{2/3}}$, which is $\frac{1}{12}$ at $x = 8$. Likewise $x\sqrt{x} = x^{3/2}$ has derivative $\frac{3}{2}x^{1/2} = \frac{3}{2}\sqrt{x}$.

So before using the rule, write every root and reciprocal as a power, using lesson 1.4.1.

Negative inputs

The slope formula works on both sides of 0, and the new power $n - 1$ decides the sign there. For $x^3$ the slope $3x^2$ has an even power, so it is positive on both sides. At $-2$ it is 12: $x^3$ rises there too.

For $x^4$ the slope $4x^3$ has an odd power, so it changes sign at 0: it is $-4$ at $x = -1$ and 4 at $x = 1$. The curve falls on the left of 0 and rises on the right.

Where the power rule breaks

The slope formula $nx^{n-1}$ shows where a power has no slope. When $n - 1$ is negative, $x$ ends up on the bottom of a fraction, and the formula fails at 0.

• Negative powers at 0. $\frac{1}{x}$ and $\frac{1}{x^2}$ have no value at $x = 0$, so they have no slope there.
• Powers between 0 and 1, at 0. $\sqrt{0} = 0$, but the slope $\frac{1}{2\sqrt{x}}$ has 0 on the bottom. Near 0 it grows without limit: 0.5 at $x = 1$, 5 at $x = 0.01$, 50 at $x = 0.0001$. The cube root's slope $\frac{1}{3x^{2/3}}$ grows the same way, to about 33.33 at $x = 0.001$.

The domain matters too. $\sqrt{x}$ has no real value for $x < 0$ (lesson 1.4.1), so there is nothing to differentiate there.

A slope that grows without limit is the vertical tangent of lesson 2.2.2. The power rule predicts it from the formula, without computing any chords.

When $n \ge 1$, as for $x^2$ or $x^3$, the slope formula has no $x$ on the bottom, and nothing breaks at 0.

Constants

A constant multiplier stays. Multiply a function by $c$ and every height is multiplied by $c$. Each rise is multiplied by $c$ and each run is not, so every slope is multiplied by $c$.

So $\frac{d}{dx}(3x^2) = 3 \times 2x = 6x$. A negative multiplier flips the graph, and the sign of every slope: $\frac{d}{dx}(-x^2) = -2x$.

An added constant vanishes. Adding a constant moves the whole graph up or down. Every rise and every run stay the same, so every slope stays the same: $\frac{d}{dx}(x^2 + 5) = 2x$. A constant on its own, like $y = 7$, is a flat line with slope 0.

A constant multiplier often carries units. A stone dropped from rest falls about $s = 4.9t^2$ metres in $t$ seconds. Its speed is $\frac{ds}{dt} = 4.9 \times 2t = 9.8t$ metres per second: 9.8 after 1 second and 19.6 after 2.

Sums and differences

When two functions are added, the rise of the sum is the sum of their rises. So the slope of a sum is the sum of the slopes.

Take $4x^3 - 2x^2 + 5x - 9$. Term by term, its derivative is $12x^2 - 4x + 5$, and the $-9$ disappears. At $x = 1$ the slope is $12 - 4 + 5 = 13$; a central difference with $h = 0.01$ gives 13.0004.

Rewrite first

Many functions become sums of powers after a little algebra. Then the rules above finish the job.

• Expand a product. $(x + 1)^2 = x^2 + 2x + 1$, so its derivative is $2x + 2$.
• Split a fraction whose bottom is a single power. $\frac{x^2 + 1}{x} = x + x^{-1}$, so its derivative is $1 - \frac{1}{x^2}$.

Two things these rules cannot do. A product is not differentiated factor by factor: $x^2 \cdot x^3 = x^5$ has slope $5x^4$, not $2x \cdot 3x^2 = 6x^3$. Lesson 2.3.2 gives the product rule.

The power rule also needs a fixed power of $x$ itself. In $2^x$ the variable is the exponent (lesson 2.3.4), and $(3x + 1)^5$ is a power of an expression (lesson 2.3.3, the chain rule).

Lesson 2.4.1 uses these rules to find where a function peaks. Case study 2.3.7 applies the power rule to every payment of a bond.

Common mistakes

1. The power drops by one. $x^5$ gives $5x^4$, not $5x^5$.
2. Negative powers move away from zero. $x^{-2}$ gives $-2x^{-3}$, not $-2x^{-1}$.
3. Rewrite as a power first. $\frac{1}{x^2}$ does not give $\frac{1}{2x}$; write it as $x^{-2}$ and get $-\frac{2}{x^3}$.
4. A constant on its own vanishes; a multiplier stays. $x^2 + 5$ gives $2x$, but $5x^2$ gives $10x$.
5. No slope where the formula breaks. $\frac{1}{x}$ has none at 0, and neither does $\sqrt{x}$, whose tangent there is vertical.
6. Not for products or variable powers. Do not differentiate a product factor by factor, and do not use the power rule on $2^x$.