The Constant Multiple and Sum Rules

A constant multiple

A number multiplying a function is called a constant multiple, or a coefficient. The constant multiple rule says: differentiate the function, and keep the number as a multiplier. So $5x^{3}$ differentiates to $5 \times 3x^{2} = 15x^{2}$. In symbols, the derivative of kf(x) is kf′(x).

From first principles, the rise of $5x^{3}$ over a step h is $5\left(x + h\right)^{3} - 5x^{3} = 5\left(\left(x + h\right)^{3} - x^{3}\right)$, five times the rise of $x^{3}$. Dividing by h keeps the 5 outside, and the limit of 5 times a quantity is 5 times its limit. Check at x = 1, where $15x^{2} = 15$: the chord with h = 0.001 gives $\left(5 \times 1.001^{3} - 5\right) / 0.001 = 15.015005$.

On a graph, $y = 3x^{2}$ is $y = x^{2}$ stretched upward by 3. Every rise is 3 times as big and every run is the same, so every gradient is 3 times as big.

The gold curve is $y = 3x^{2}$, and the plain curve is $y = x^{2}$. At x = 1 the tangent to $y = x^{2}$ has gradient 2, and the tangent to $y = 3x^{2}$ has gradient 3 × 2 = 6.

A sum, term by term

The sum rule says: to differentiate a sum, differentiate each term on its own and add the results. The derivative of f(x) + g(x) is f′(x) + g′(x). A difference works the same way, with a minus sign.

The reason is the difference quotient. The rise of f + g over a step h is the rise of f plus the rise of g, so the quotient splits into two quotients, and the limit of a sum is the sum of the limits.

So $x^{3} + 4x^{2} - 7$ differentiates term by term: $x^{3}$ gives $3x^{2}$, $4x^{2}$ gives 8x, and −7 gives 0. The derivative is $3x^{2} + 8x$. At x = 1 that is 3 + 8 = 11, and the chord with h = 0.001 gives 11.007001.

The gold curve is $y = x^{2} + x$, and the plain curves are its two terms, $y = x^{2}$ and the line y = x. At x = 1, $x^{2}$ has gradient 2 and x has gradient 1, and the tangent to the sum at (1, 2) has gradient 2 + 1 = 3.

A constant term

A constant term such as −7 differentiates to 0. Its graph, y = −7, is a level line with gradient 0 everywhere.

Adding or taking away a constant moves a whole curve up or down without tilting any part of it. The curves $y = x^{2}$ and $y = x^{2} - 3$ have the same gradient, 2x, at every x: at x = 1 both tangents have gradient 2. So $x^{3} + 4x^{2} - 7$ and $x^{3} + 4x^{2}$ have the same derivative, $3x^{2} + 8x$.

The gold curve is $y = x^{2} - 3$, and the plain curve is $y = x^{2}$, 3 higher at every x. Their tangents at x = 1 are parallel, both with gradient 2.

Putting them together

With the power rule, these two rules differentiate any sum of powers of x. $4x^{3} - 7x + 2$ gives $4 \times 3x^{2} - 7 \times 1 + 0 = 12x^{2} - 7$. At x = 1 that is 5, and the chord with h = 0.001 gives 5.012004.

Rewrite roots and reciprocals as powers first. $2x^{5} - \frac{3}{x} + 6\sqrt{x}$ is $2x^{5} - 3x^{-1} + 6x^{\frac{1}{2}}$. Term by term: $2x^{5}$ gives $10x^{4}$; $-3x^{-1}$ gives $-3 \times \left(-x^{-2}\right) = 3x^{-2}$; $6x^{\frac{1}{2}}$ gives $6 \times \left(\frac{1}{2}\right)x^{- \frac{1}{2}} = 3x^{- \frac{1}{2}}$. So the derivative is $10x^{4} + \frac{3}{x^{2}} + \frac{3}{\sqrt{x}}$, which is 10 + 3 + 3 = 16 at x = 1. The chord with h = 0.0001 gives 16.001625.

A fraction with a single term underneath can be divided term by term first. $\frac{{x^{2} + 3}}{x} = x + 3x^{-1}$, which gives $1 - 3x^{-2} = 1 - \frac{3}{x^{2}}$. At x = 1 that is −2, and the chord with h = 0.001 gives −1.997003.

A product of two brackets is not a sum, so these rules do not apply to it as it stands. Expand it first: $\left(x + 2\right)\left(x - 3\right) = x^{2} - x - 6$, which gives 2x − 1. At x = 2 that is 3, and the chord with h = 0.001 gives 3.001. Differentiating each bracket and multiplying, 1 × 1 = 1, is wrong.

The usual mistakes

Keeping the constant term: $3x^{2} + 5x + 4$ to 6x + 5 + 4. The + 4 has gradient 0, so the derivative is 6x + 5.

Dropping the power without multiplying by it: $3x^{2}$ to 3x. The power rule turns $x^{2}$ into 2x, so $3x^{2}$ gives 3 × 2x = 6x.

Treating a coefficient like a constant term. The 5 in $5x^{3}$ multiplies $x^{3}$ and stays: the derivative is $15x^{2}$, not $3x^{2}$ and not 0. Only a number standing alone, added or taken away, differentiates to 0.

Losing a minus sign. $- \frac{3}{x}$ is $-3x^{-1}$, and the power rule brings down another −1, so the derivative is $+ \frac{3}{x^{2}}$.

A tram

In the application below, a tram's distance from its stop is $s = 0.4x^{2} + 0.02x^{3}$ meters after x seconds. Each differentiation is term by term, with each coefficient kept: once for the velocity, twice for the acceleration, and three times for the rate the acceleration changes.