Constant Multiple Rule for finding Derivatives

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:

ddx(cf(x))=c·ddx(f(x))

where c is a constant and f(x) is a function of x.

For a function g(x)=c·f(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(x)=ddx(c·f(x))=c·ddx(f(x))=c·f(x)

Intuitive Understanding of the Constant Multiple Rule

To understand the Constant Multiple Rule intuitively, consider the following example. Let f(x)=x2 (plotted in orange below), and let’s multiply this function by a constant c=3 to get c·g(x)=3·(x2)=3x2 (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

  1. Identify the constant coefficient: Determine the constant c that is multiplying the function f(x).

  2. Find the derivative of the inner function: Calculate ddx(f(x)), which is the derivative of the function being multiplied by the constant.

  3. Multiply the constant and the derivative: Multiply the constant c from step 1 and the derivative ddx(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(x)=limh0g(x+h)g(x)h

Step 1: Substitute g(x)=cf(x) into the definition of the derivative.

g(x)=limh0c·f(x+h)c·f(x)h

Step 2: Factor out the constant c.

g(x)=c·limh0f(x+h)f(x)h

Step 3: Recognize that limh0f(x+h)f(x)h=f(x), which is the definition of the derivative of f(x).

g(x)=c·f(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

  1. Find the derivative of f(x)=3x2+5x.

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

    f(x)=3·ddx(x2)+5·ddx(x)=3·2x+5·1=6x+5
  2. Find the derivative of g(x)=2sin(x).

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

    g(x)=2·ddx(sin(x))=2·cos(x)
  3. A particle’s position is given by the function s(t)=4t32t, 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(t)=4·ddt(t3)2·ddt(t)=4·3t22·1=12t22

    To find the acceleration, we take the derivative of the velocity function:

    a(t)=v(t)=12·ddt(t2)=12·2t=24t

    So, the particle’s velocity is v(t)=12t22 meters per second, and its acceleration is a(t)=24t meters per second squared.

  4. 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(x)=100·ddx(x)+500·ddx(1)=100·1+500·0=100

    So, the marginal cost is constant at 100 dollars per unit.

  5. Solve the differential equation y4y+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 with r2, y with r, and y with 1:

    r24r+4=0

    Factoring this equation, we get:

    (r2)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)=(C1+C2x)e2x

    where C1 and C2 are arbitrary constants determined by initial conditions.

    Note that the Constant Multiple Rule was used implicitly when we replaced y with r2, y with r, and y with 1 in the characteristic equation, as this is equivalent to finding the derivatives of erx and multiplying by the constant coefficients.