The **Constant Multiple Rule,** also known as the *Constant Coefficient Rule,* is a rule in calculus used for differentiating functions that are multiplied by a **constant.** This rule simplifies the process of finding derivatives for functions involving constant multiples. The Constant Multiple Rule states that the derivative of a constant times a function is equal to the constant times the derivative of the function.

## Constant Multiple Rule Formula and Formal Definition

The Constant Multiple Rule formula is as follows:

$$\frac{\text{d}}{\text{d}x}(cf(x))=c\xb7\frac{\text{d}}{\text{d}x}(f(x))$$where $c$ is a constant and $f(x)$ is a function of $x$.

For a function $g(x)=c\xb7f(x)$, where $c$ is a constant and $f(x)$ is a function of $x$, the derivative of $g(x)$ with respect to $x$ is given by:

$${g}^{\prime}(x)=\frac{\text{d}}{\text{d}x}(c\xb7f(x))=c\xb7\frac{\text{d}}{\text{d}x}(f(x))=c\xb7{f}^{\prime}(x)$$## Intuitive Understanding of the Constant Multiple Rule

To understand the Constant Multiple Rule intuitively, consider the following example. Let $f(x)={x}^{2}$ (plotted in orange below), and let’s multiply this function by a constant $c=3$ to get $c\xb7g(x)=3\xb7({x}^{2})=3{x}^{2}$ (in blue below).

If we think about the graph of $g(x)$, it will have the same shape as the graph of $f(x)$, but it will be vertically stretched by a factor of 3. This means that for any change in $x$, the change in $g(x)$ will be 3 times the change in $f(x)$.

Now, recall that the derivative of a function at a point is the slope of the tangent line to the function’s graph at that point. Since the graph of $g(x)$ is stretched vertically by a factor of 3, the slope of the tangent line to $g(x)$ at any point will be 3 times the slope of the tangent line to $f(x)$ at the corresponding point.

Therefore, the derivative of $g(x)$ will be 3 times the derivative of $f(x)$, which is exactly what the Constant Multiple Rule states.

## Steps to Apply the Constant Multiple Rule

**Identify the constant coefficient:**Determine the constant $c$ that is multiplying the function $f(x)$.**Find the derivative of the inner function:**Calculate $\frac{\text{d}}{\text{d}x}(f(x))$, which is the derivative of the function being multiplied by the constant.**Multiply the constant and the derivative:**Multiply the constant $c$ from step 1 and the derivative $\frac{\text{d}}{\text{d}x}(f(x))$ from step 2 to obtain the final result.

## Proof of the Constant Multiple Rule

To prove the Constant Multiple Rule, we can use the definition of the derivative:

$${g}^{\prime}(x)={lim}_{h\to 0}\frac{g(x+h)-g(x)}{h}$$Step 1: Substitute $g(x)=cf(x)$ into the definition of the derivative.

$${g}^{\prime}(x)={lim}_{h\to 0}\frac{c\xb7f(x+h)-c\xb7f(x)}{h}$$Step 2: Factor out the constant $c$.

$${g}^{\prime}(x)=c\xb7{lim}_{h\to 0}\frac{f(x+h)-f(x)}{h}$$Step 3: Recognize that ${lim}_{h\to 0}\frac{f(x+h)-f(x)}{h}={f}^{\prime}(x)$, which is the definition of the derivative of $f(x)$.

$${g}^{\prime}(x)=c\xb7{f}^{\prime}(x)$$Thus, we have proven that the derivative of a constant times a function is equal to the constant times the derivative of the function.

## Examples

Find the derivative of $f(x)=3{x}^{2}+5x$.

Using the Constant Multiple Rule and the Power Rule, we get:

$${f}^{\prime}(x)=3\xb7\frac{\text{d}}{\text{d}x}({x}^{2})+5\xb7\frac{\text{d}}{\text{d}x}(x)=3\xb72x+5\xb71=6x+5$$Find the derivative of $g(x)=-2\mathrm{sin}(x)$.

Using the Constant Multiple Rule and the derivative of sine, we get:

$${g}^{\prime}(x)=-2\xb7\frac{\text{d}}{\text{d}x}(\mathrm{sin}(x))=-2\xb7\mathrm{cos}(x)$$A particle’s position is given by the function $s(t)=4{t}^{3}-2t$, where $s$ is measured in meters and $t$ is measured in seconds. Find the particle’s velocity and acceleration at time $t$.

To find the velocity, we take the derivative of the position function using the Constant Multiple Rule and the Power Rule:

$$v(t)={s}^{\prime}(t)=4\xb7\frac{\text{d}}{\text{d}t}({t}^{3})-2\xb7\frac{\text{d}}{\text{d}t}(t)=4\xb73{t}^{2}-2\xb71=12{t}^{2}-2$$To find the acceleration, we take the derivative of the velocity function:

$$a(t)={v}^{\prime}(t)=12\xb7\frac{\text{d}}{\text{d}t}({t}^{2})=12\xb72t=24t$$So, the particle’s velocity is $v(t)=12{t}^{2}-2$ meters per second, and its acceleration is $a(t)=24t$ meters per second squared.

Find the marginal cost function if the total cost function is given by $C(x)=100x+500$, where $C(x)$ is the total cost in dollars and $x$ is the number of units produced.

The marginal cost function is the derivative of the total cost function. Using the Constant Multiple Rule, we get:

$$MC(x)={C}^{\prime}(x)=100\xb7\frac{\text{d}}{\text{d}x}(x)+500\xb7\frac{\text{d}}{\text{d}x}(1)=100\xb71+500\xb70=100$$So, the marginal cost is constant at 100 dollars per unit.

Solve the differential equation ${y}^{\u2033}-4{y}^{\prime}+4y=0$.

This is a second-order linear differential equation with constant coefficients. To solve it, we first find the characteristic equation by replacing ${y}^{\u2033}$ with ${r}^{2}$, ${y}^{\prime}$ with $r$, and $y$ with $1$:

$${r}^{2}-4r+4=0$$Factoring this equation, we get:

$$(r-2{)}^{2}=0$$So, the characteristic equation has a double root at $r=2$. This means that the general solution to the differential equation is:

$$y(x)=({C}_{1}+{C}_{2}x){e}^{2x}$$where ${C}_{1}$ and ${C}_{2}$ are arbitrary constants determined by initial conditions.

Note that the Constant Multiple Rule was used implicitly when we replaced ${y}^{\u2033}$ with ${r}^{2}$, ${y}^{\prime}$ with $r$, and $y$ with $1$ in the characteristic equation, as this is equivalent to finding the derivatives of ${e}^{rx}$ and multiplying by the constant coefficients.