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

Finding the derivative of $p(x) = 4x^3 - 7x + 2$ from the limit definition means expanding $(x+h)^3$, cancelling, and dividing by $h$, and the next polynomial would need the same work again. Instead, you can differentiate $4x^3$, $-7x$ and $2$ separately and combine the results. Three rules make that legal: a constant has derivative 0, a constant multiplier passes through the derivative, and the derivative of a sum or difference is the sum or difference of the derivatives. Each is a short consequence of the limit definition, and together with the power rule (topic 2.5) they differentiate every polynomial.

A constant has derivative zero

The graph of a constant function $f(x) = c$ is a horizontal line, and a horizontal line has slope 0 everywhere. The limit definition says the same thing: $f(x+h)$ and $f(x)$ are both $c$, so every difference quotient is

and the limit as $h \to 0$ is 0.

"Constant" means anything that does not depend on $x$, however it is written. The numbers $\pi^2$, $e^3$, $\ln 5$ and $\sqrt{17}$ are all constants, so $\frac{d}{dx}\left(\pi^2\right) = 0$. A letter that names a fixed parameter, such as $k$ in $y = x^2 + k$, is a constant too.

A constant multiplier passes through

If $g(x) = k\,f(x)$ for a constant $k$, every output of $f$ is multiplied by $k$, so every rise on the graph is multiplied by $k$ while every run stays the same. Every slope is therefore multiplied by $k$. In the limit definition, the $k$ factors out of the difference quotient,

and the limit of a constant times something is the constant times the limit (the limit laws from topic 1.5). So $g'(x) = k\,f'(x)$, provided $f$ is differentiable at $x$.

A negative $k$ flips the graph upside down, and the slopes flip sign with it: if $f$ is rising with slope 4 at some point, $-3f$ is falling with slope $-12$ there.

Compare the two rules so far. In $5x^4$ the 5 multiplies a function of $x$, so it stays: $\frac{d}{dx}\left(5x^4\right) = 5\cdot 4x^3 = 20x^3$. In $x^4 + 5$ the 5 stands alone, so it disappears: $\frac{d}{dx}\left(x^4 + 5\right) = 4x^3$. Adding a constant shifts the graph up or down, which changes no slope; multiplying by a constant stretches the graph, which changes every slope.

Sums and differences go term by term

If $s(x) = f(x) + g(x)$, then a change in $x$ changes $s$ by the change in $f$ plus the change in $g$. Dividing by $h$ splits the difference quotient into two:

When $f$ and $g$ are both differentiable at $x$, each fraction on the right has a limit, and the limit of a sum is the sum of the limits. The same argument with a minus sign handles differences.

The hypothesis matters in one direction only. If both pieces are differentiable, so is the sum. If one piece is not, the rule tells you nothing, and you have to look at the whole function. For example, $|x|$ is not differentiable at 0, yet $|x| + (-|x|)$ is the constant function 0, which is differentiable everywhere.

These rules also handle a function you know only by name. If $f'(2) = 3$ and $g'(2) = -1$, then the derivative of $4f(x) - g(x)$ at $x = 2$ is $4(3) - (-1) = 13$, with no formula for $f$ or $g$ needed.

Polynomials and other sums of powers

The power rule from topic 2.5 says $\frac{d}{dx}x^n = nx^{n-1}$ for any constant exponent $n$. A polynomial is a sum of constant multiples of powers, so the three rules let you apply the power rule one term at a time:

Each term drops one degree, the linear term $-7x$ becomes the constant $-7$, and the constant term vanishes. So for $n \ge 1$, the derivative of a polynomial of degree $n$ is a polynomial of degree $n - 1$. A constant polynomial (degree 0) is the exception: its derivative is 0 everywhere.

The rules in this topic apply only to sums, differences and constant multiples. They do not cover a product or quotient of two functions of $x$; topics 2.8 and 2.9 handle those. Many products and quotients can be rewritten as sums of powers first, though, and that rewrite is often the fastest route:

Two details in that table matter. In the second row, the original function is undefined at $x = 0$ even after you cancel, so its derivative is undefined there too. In the fourth row, the 4 sits in the denominator with $x^3$, so the coefficient is $\frac{1}{4}$, not 4.

Worked examples

Where students lose points

On the exam

These rules show up inside tangent-line problems, motion problems and optimization. On the no-calculator section you are expected to differentiate polynomials and sums of powers by hand. When a function is given only through values of $f$ and $f'$, the constant multiple and sum rules are how you find the derivative of a combination such as $3f(x) - g(x)$.