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$5x³ differentiates to $5\times 3x^2=15x^2$5 × 3x² = 15x². In symbols, the derivative of kf(x) is kf′(x).

From first principles, the rise of $5x^3$5x³ over a step h is $5{\left(x+h\right)}^3-5x^3=5\left({\left(x+h\right)}^3-x^3\right)$5(x + h)³ − 5x³ = 5((x + h)³ − x³), five times the rise of $x^3$x³. 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$15x² = 15: the chord with h = 0.001 gives $\left(5\times 1.001^3-5\right)/0.001=15.015005$(5 × 1.001³ − 5) / 0.001 = 15.015005.

On a graph, $y=3x^2$y = 3x² is $y=x^2$y = x² 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 $y=3x^2$y = 3x², and the plain curve is $y=x^2$y = x². At x = 1 the tangent to $y=x^2$y = x² has gradient 2, and the tangent to $y=3x^2$y = 3x² 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 $x^3+4x^2-7$x³ + 4x² − 7 differentiates term by term: $x^3$x³ gives $3x^2$3x², $4x^2$4x² gives 8x, and −7 gives 0. The derivative is $3x^2+8x$3x² + 8x. At x = 1 that is 3 + 8 = 11, and the chord with h = 0.001 gives 11.007001.

The gold curve is $y=x^2+x$y = x² + x, and the plain curves are its two terms, $y=x^2$y = x² and the line y = x. At x = 1, $x^2$x² 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 $y=x^2$y = x² and $y=x^2-3$y = x² − 3 have the same gradient, 2x, at every x: at x = 1 both tangents have gradient 2. So $x^3+4x^2-7$x³ + 4x² − 7 and $x^3+4x^2$x³ + 4x² have the same derivative, $3x^2+8x$3x² + 8x.

The gold curve is $y=x^2-3$y = x² − 3, and the plain curve is $y=x^2$y = x², 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$4x³ − 7x + 2 gives $4\times 3x^2-7\times 1+0=12x^2-7$4 × 3x² − 7 × 1 + 0 = 12x² − 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}$2x⁵ − 3/x + 6√x is $2x^5-3x^{-1}+6x^{\frac{1}{2}}$2x⁵ − 3x⁻¹ + 6x^(1/2). Term by term: $2x^5$2x⁵ gives $10x^4$10x⁴; $-3x^{-1}$−3x⁻¹ gives $-3\times \left(-x^{-2}\right)=3x^{-2}$−3 × (−x⁻²) = 3x⁻²; $6x^{\frac{1}{2}}$6x^(1/2) gives $6\times \left(\frac{1}{2}\right)x^{-\frac{1}{2}}=3x^{-\frac{1}{2}}$6 × (1/2)x^(−1/2) = 3x^(−1/2). So the derivative is $10x^4+\frac{3}{x^2}+\frac{3}{\sqrt{x}}$10x⁴ + 3/x² + 3/√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. $\frac{x^2+3}{x}=x+3x^{-1}$(x² + 3)/x = x + 3x⁻¹, which gives $1-3x^{-2}=1-\frac{3}{x^2}$1 − 3x⁻² = 1 − 3/x². 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: $\left(x+2\right)\left(x-3\right)=x^2-x-6$(x + 2)(x − 3) = x² − 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$3x² + 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$3x² to 3x. The power rule turns $x^2$x² into 2x, so $3x^2$3x² gives 3 × 2x = 6x.

Treating a coefficient like a constant term. The 5 in $5x^3$5x³ multiplies $x^3$x³ and stays: the derivative is $15x^2$15x², not $3x^2$3x² and not 0. Only a number standing alone, added or taken away, differentiates to 0.

Losing a minus sign. $-\frac{3}{x}$−3/x is $-3x^{-1}$−3x⁻¹, and the power rule brings down another −1, so the derivative is $+\frac{3}{x^2}$+3/x².

A tram

In the application below, a tram's distance from its stop is $s=0.4x^2+0.02x^3$s = 0.4x² + 0.02x³ 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.