2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

The power rule of Topic 2.5, $\frac{d}{dx}(x^r)=rx^{r-1}$, differentiates a single power of $x$. Most functions are built from several such pieces, scaled by coefficients, added, subtracted and shifted by constants. Four structural rules describe how a derivative passes through that algebra, so a polynomial can be differentiated one term at a time.

Learning Objective FUN-3.A: Calculate derivatives of familiar functions.
Essential Knowledge FUN-3.A.2: Sums, differences, and constant multiples of functions can be differentiated using derivative rules.
Essential Knowledge FUN-3.A.3: The power rule combined with sum, difference, and constant multiple properties can be used to find the derivatives for polynomial functions.

The four structural rules

A constant function has the same output for every input. Its graph is horizontal, so its instantaneous rate of change is zero:

$$ \frac{d}{dx}(C)=0, $$

where $C$ is any constant.

The sum rule differentiates each function separately:

$$ \frac{d}{dx}\bigl(f(x)+g(x)\bigr) =f'(x)+g'(x). $$

The difference rule works the same way:

$$ \frac{d}{dx}\bigl(f(x)-g(x)\bigr) =f'(x)-g'(x). $$

The constant multiple rule leaves a constant factor outside the derivative:

$$ \frac{d}{dx}\bigl(Cf(x)\bigr)=Cf'(x). $$

These rules are best viewed as a “preserve the structure” system:

Function structure	Derivative action
Constant $C$	Replace it with $0$
Sum $f+g$	Differentiate both terms and add
Difference $f-g$	Differentiate both terms and subtract
Constant multiple $Cf$	Keep $C$ and differentiate $f$

Worked example: a changing production rate

Suppose the number of components produced by a machine after $t$ hours is modeled by

$$ P(t)=4t^5-9t^3+12t-80. $$

The instantaneous production rate is $P'(t)$. Differentiate term by term:

$$ \begin{aligned} P'(t) &=\frac{d}{dt}(4t^5) -\frac{d}{dt}(9t^3) +\frac{d}{dt}(12t) -\frac{d}{dt}(80)\[4pt] &=4(5t^4)-9(3t^2)+12(1)-0\[4pt] &=20t^4-27t^2+12. \end{aligned} $$

Thus,

$$ \boxed{P'(t)=20t^4-27t^2+12}. $$

At $t=2$ hours,

$$ P'(2)=20(2^4)-27(2^2)+12 =320-108+12 =224. $$

So the model predicts an instantaneous production rate of $224$ components per hour at $t=2$.

Why term-by-term differentiation works

The derivative is built from a limit, and limits preserve sums, differences, and constant factors. That is why a polynomial does not require a new technique for every term: each term contributes its own rate of change, and those contributions combine according to the original algebraic signs and coefficients.

For instance, if

$$ f(x)=x^4-6x^2+3, $$

then

$$ f'(x)=4x^3-12x. $$

The constant $3$ contributes no change, the subtraction remains subtraction, and the coefficient $6$ remains attached to the derivative of $x^2$.

Misconception check

Misconception: “The derivative of $7x^4$ is $4x^3$.”

The exponent rule gives the derivative of $x^4$, not of the whole term. The coefficient must remain:

$$ \frac{d}{dx}(7x^4)=7(4x^3)=28x^3. $$

A second common error is differentiating a constant as though it were a variable expression:

$$ \frac{d}{dx}(11)=0, $$

not $1$ and not $11$.

A third is mishandling subtraction. The negative sign belongs to the term:

$$ \frac{d}{dx}\left(x^5-3x^2\right) =5x^4-6x, $$

not $5x^4+6x$.

Using procedures with and without technology

For a familiar polynomial, an analytical derivative is usually fastest and most informative. A graphing calculator or computer algebra system can verify the result, but technology should not replace the rule selection or the written derivative.

For

$$ h(x)=2x^3-5x+9, $$

the analytical calculation is

$$ h'(x)=6x^2-5. $$

A calculator can numerically estimate $h'(2)$, but the exact derivative immediately gives

$$ h'(2)=6(2^2)-5=19. $$

This is an instance of Suggested Skill 1.E: Apply appropriate mathematical rules or procedures, with and without technology.

Retrieval check

Differentiate

$$ q(x)=6x^4-2x^3+5x-14. $$

Apply the power rule, sum and difference rules, constant multiple rule, and constant-function rule:

$$ \boxed{q'(x)=24x^3-6x^2+5}. $$

If a coefficient or sign changes in your answer, identify the original term that caused it before continuing.