Constant multiple rule · Derivative Calculator

What it says

Scaling a function by a fixed number scales its rate of change by that same number. Stretch a graph vertically by $3$ and every slope on it triples.

This is why coefficients can be pulled out of a derivative and dealt with at the end. Together with the sum rule it gives linearity: $\left(af + bg\right)' = af' + bg'$, the property that makes differentiation well behaved on polynomials.

Division by a constant is the same rule in disguise, since $\frac{f}{c}$ is $\frac{1}{c}f$. Reaching for the quotient rule there is legal but wasteful.

When it applies

A number or a constant symbol multiplies a function of $x$.
A function is divided by a constant.
A constant is buried inside a product, such as the $\cos 9$ in $6x\cos(9)\sin x$ — it is just a number.

Five worked examples

Every line is the step the calculator would show, in the order it applies them. Each graph is live: hover it to read both curves and see the tangent whose slope is the derivative, drag to pan, scroll to zoom.

Example 1

\frac{d}{dx}\left[7 x^{3}\right]

Pull the $7$ out.
$\frac{d}{dx}\left[7x^{3}\right] = 7\;\frac{d}{dx}\left[x^{3}\right]$
Power rule, then multiply back.
$= 7 \cdot 3x^{2} = 21x^{2}$

Answer

21 x^{2}

A steepened cubic with the narrow parabola $21x^{2}$ beneath it. f′ touches zero at the origin, where the cubic momentarily flattens, and is positive on both sides. Multiplying by 7 stretched both curves vertically by the same factor.

The coefficient is never differentiated; it simply multiplies the result.

Open 7x^3 in the calculator

Example 2

\frac{d}{dx}\left[-4 \sin\left(x\right)\right]

The constant is $-4$, sign included.
$= -4\;\frac{d}{dx}\left[\sin x\right]$
Trigonometric rule.
$= -4\cos x$

Answer

-4 \cos\left(x\right)

Scaling by $-4$ flips the wave and its slope together, so f′ peaks on the opposite side of the axis from the unscaled case.

Keep the sign attached to the constant rather than juggling it separately.

Open -4 sin(x) in the calculator

Example 3

\frac{d}{dx}\left[\frac{x^{2}}{5}\right]

Dividing by $5$ is multiplying by $\tfrac{1}{5}$.
$\frac{x^{2}}{5} = \tfrac{1}{5}x^{2}$
Constant multiple, then power rule.
$= \tfrac{1}{5}\cdot 2x = \frac{2x}{5}$

Answer

\frac{2 x}{5}

A parabola flattened to a fifth of its usual height, with the line $\tfrac{2x}{5}$ for its slope — a fifth as steep as the usual $2x$.

No quotient rule needed — the denominator holds no $x$.

Open x^2/5 in the calculator

Example 4

\frac{d}{dx}\left[3 \ln\left(x\right)\right]

Take the $3$ outside.
$= 3\;\frac{d}{dx}\left[\ln x\right]$
Logarithm rule.
$= 3 \cdot \frac{1}{x} = \frac{3}{x}$

Answer

\frac{3}{x}

f rises ever more slowly while f′ decays toward zero: the familiar $\tfrac{1}{x}$ curve, three times as tall.

Equivalently $3\ln x = \ln x^{3}$, which differentiates to $\frac{3x^{2}}{x^{3}} = \frac{3}{x}$ — the same answer by a longer road.

Open 3 ln(x) in the calculator

Example 5

\frac{d}{dx}\left[6 \cos\left(9\right) \sin\left(x\right)\right]

$\cos 9$ has no $x$ in it, so $6\cos 9$ is a single constant factor.
$= 6\cos(9)\;\frac{d}{dx}\left[\sin x\right]$
Only $\sin x$ is differentiated.
$= 6\cos(9)\cos x$

Answer

6 \cos\left(9\right) \cos\left(x\right)

$\cos 9 \approx -0.911$, so the whole constant is about $-5.47$ and f is an upside-down sine of that amplitude.

A trigonometric function of a number is a number. Scan for the variable before deciding what is constant.

Open 6 cos(9) sin(x) in the calculator

Watch out for

Only constants come out. $\frac{d}{dx}\left[x\sin x\right] \neq x\cos x$ — that needs the product rule.
The constant is not sent to zero here; that is the constant rule, which applies to a standalone term.
A constant in an exponent is not a factor: $x^{3}$ needs the power rule, not this one.

Sum rule Product rule Constant rule

What it says

When it applies

Five worked examples

Watch out for

Related rules