Constant Multiple Rule

The constant multiple rule is a general rule that is used in calculus when an operation is applied on a function multiplied by a constant. We have different constant multiple rules for differentiation, limits, and integration in calculus. The general statement of the constant multiple rule is when an operation (differentiation, limits, or integration) is applied to the product of a constant and a function, then it is equal to the product of the constant and operation applied on the function.

Further, in this article, we will explore the concept of constant multiple rule of differentiation, limits and integration in detail. We will understand their formulas, proof of constant multiple rule, and its application with the help of solved examples for a better understanding.

The constant multiple rule in calculus is used for three operations, namely integration, limits, and differentiation. The general statement of the constant multiple rule for the three operations is:

• Constant Multiple Rule of Differentiation: It states that the derivative of the product of a constant with a function f(x) is equal to the product of the constant with the derivative of the function f(x).
• Constant Multiple Rule of Limits: It states that the limit of constant times a function f(x) is equal to the product of the constant with the limit of the function.
• Constant Multiple Rule of Integration: This rule states that the integral of the product of a constant with a function f(x) is equal to the product of the constant with the integral of the function f(x).

We apply these rules to simplify the mathematical problems. Let us now understand their formulas in the next section.

In this section, we will write the formulas for the constant multiple rule of limits, differentiation, and integration. Consider a constant k and function f(x), then the required formulas are as follows:

• Constant Multiple Rule of Differentiation: d(k f(x))/dx = k d(f((x))/dx
• Constant Multiple Rule of Limits: lim_x→a k f(x) = k lim_x→a f(x)
• Constant Multiple Rule of Integration: ∫k f(x) dx = k ∫f(x) dx

Now that we know the constant multiple rule formulas for differentiation, integration, and limits, we will prove those formulas using different formulas and properties of derivatives, integrals, and limits. Let us first derive the formula for the constant multiple rule of differentiation.

Constant Multiple Rule of Differentiation

The constant multiple rule of derivatives states that the derivative of the product of a constant with a function f(x) is equal to the product of the constant with the derivative of the function f(x). To prove the formula for this, we will use the first principle of differentiation, that is, the definition of limits. We will prove the constant multiple rule formula d(k f(x))/dx = k d(f((x))/dx using the following formulas:

• f'(x) = lim_h→0 [f(x+h) - f(x)] / h --- (1)
• Product Rule of Limits: lim_x→a [ f(x) × g(x) ] = lim_x→a f(x) × lim_x→a g(x)
• Limit of a constant: lim_x→a k = k

Assume h(x) = k f(x). Using the above formulas, we have

LHS = h'(x)

= d(k f(x))/dx

= lim_h→0 [k f(x+h) - k f(x)] / h

= lim_h→0 {k [f(x+h) - f(x)] / h } --- [Taking the common factor k out]

= lim_h→0 {k × [f(x+h) - f(x)] / h }

= lim_h→0 k × lim_h→0 [f(x+h) - f(x)] / h

= k × lim_h→0 [f(x+h) - f(x)] / h --- [Using the limit of a constant formula lim_x→a k = k]

= k × f'(x) --- [Using (1)]

= k d[f(x)]/dx

= RHS

Hence, we have proved the constant multiple rule of differentiation.

Constant Multiple Rule of Limits

To prove the formula for the constant multiple rule for limits given by lim_x→a k f(x) = k lim_x→a f(x), we will use the formula for the limit of the product of functions, and the formula for a constant function given by,

• Product Rule of Limits: lim_x→a [ f(x) × g(x) ] = lim_x→a f(x) × lim_x→a g(x)
• Limit of a constant: lim_x→a k = k

Using the above formulas, we have

LHS = lim_x→a k f(x)

= lim_x→a [k × f(x)]

= lim_x→a k × lim_x→a f(x) --- [Using product rule of limits]

= k × lim_x→a f(x) --- [Limit of a constant is equal to the constant itself]

= k lim_x→a f(x)

= RHS

Hence, we have proved the formula for the constant multiple rule for limits.

Constant Multiple Rule for Integrals

In this section, we will prove the formula for the constant multiple rule of integration given by, ∫k f(x) dx = k ∫f(x) dx using the integration by parts method of integration. It is used to determine the integral of the product of functions. Its formula is given by, ∫[g(x) h(x)] dx = g(x) × ∫h(x) dx - ∫[g'(x) × ∫h(x) dx] dx. To prove the constant multiple rule for integrals, assume g(x) = k and h(x) = f(x). Then, we have

∫kf(x) dx = k × ∫f(x) dx - ∫[(k)' × ∫f(x) dx] dx

= k ∫f(x) dx - ∫[0 × ∫f(x) dx] dx --- [Because derivative of a constant is always equal to zero]

= k ∫f(x) dx - 0

= k ∫f(x)

= RHS

Hence, we have derived and proved the formula for the constant multiple rule of integration.

Now that we have understood the formulas and derivation of the constant multiple rule, let us solve a few examples based on the concept to understand its application. Consider a function f(x) = x² + 3x - 4 and k = 5. Now, apply the constant multiple rule formula to find the derivative, integral, and limit of k f(x).

Applying Constant Multiple Rule for Derivatives

Now, we will use the constant multiple rule to find the derivative of kf(x) = 5 [x² + 3x - 4] = 5x² + 15x - 20 an verify the result using the power rule. According to the constant multiple rule, we have

d(kf(x))/dx = d[5 (x² + 3x - 4)]/dx

= 5 d[x² + 3x - 4]/dx --- [Using constant multiple rule]

= 5 × (2x + 3) --- [Using Power Rule]

= 10x + 15

To verify the result, let us differentiate kf(x) = 5x² + 15x - 20 using the power rule of differentiation:

Verification: d(kf(x))/dx

= d(5x² + 15x - 20)/dx

= 10x + 15

Hence, we have applied the constant multiple rule of differentiation and verified the result.

Applying Constant Multiple Rule of Limits

Now, we will find the limit of the function kf(x) = 5 [x² + 3x - 4], where k = 5 and f(x) = x² + 3x - 4 as x tends to 3 using the constant multiple rule for limits and verify the result by taking the limit of the function kf(x) = 5x² + 15x - 20 with x tending towards 3. So, we have

lim_x→3 k f(x) = lim_x→3 5 [x² + 3x - 4]

= 5 lim_x→3 [x² + 3x - 4] --- [Using constant multiple rule for limits]

= 5 × (3² + 3(3) - 4)

= 5 × (9 + 9 - 4)

= 5 × 14

= 70

Now, to verify the result, we will find the limit of the function kf(x) = 5x² + 15x - 20 as x tends to 3.

Verification: lim_x→3 k f(x)

= lim_x→3 (5x² + 15x - 20)

= 5(3)² + 15(3) - 20

= 5 × 9 + 15 × 3 - 20

= 45 + 45 - 20

= 90 - 20

= 70

Hence, we have understood the application of constant multiple rule for limits.

Applying Constant Multiple Rule of Integration

In this section, we will calculate the integral of kf(x) = 5 [x² + 3x - 4] using the constant multiple rule of integrals and verify the result by calculating the integral of kf(x) = 5x² + 15x - 20 using the power rule of integration. So, we have

∫kf(x) dx = ∫5 [x² + 3x - 4] dx

= 5 ∫(x² + 3x - 4) dx

= 5 [x³/3 + 3x²/2 - 4x] + C

= 5x³/3 + 15x²/2 - 20x + C --- (1)

Now, to verify the result, we will find the integral of kf(x) = 5x² + 15x - 20 using the power rule of integration.

∫kf(x) dx = ∫[5x² + 15x - 20] dx

= ∫5x² dx + ∫15x dx - ∫20 dx

= 5x³/3 + 15x²/2 - 20x + C --- (2)

From (1) and (2), we have verified the constant multiple rule for integration.

Important Notes on Constant Multiple Rule

• The constant multiple rule in calculus is used for three operations, namely integration, limits, and differentiation.
• Constant Multiple Rule of Differentiation: d(k f(x))/dx = k d(f((x))/dx
• Constant Multiple Rule of Limits: lim_x→a k f(x) = k lim_x→a f(x)
• Constant Multiple Rule of Integration: ∫k f(x) dx = k ∫f(x) dx

☛ Related Topics:

• Integral of log x
• Derivative of xlnx
• Antilog Table

What is the Constant Multiple Rule in Calculus?

The constant multiple rule is a general rule that is used in calculus when an operation is applied on a function multiplied by a constant. The general statement of the constant multiple rule is when an operation (differentiation, limits, or integration) is applied to the product of a constant and a function, then it is equal to the product of the constant and operation applied on the function.

What are the Constant Multiple Rule Formulas?

The general formulas for the constant multiple rule for differentiation, limits and integration are:

• Constant Multiple Rule of Differentiation: d(k f(x))/dx = k d(f((x))/dx
• Constant Multiple Rule of Limits: lim_x→a k f(x) = k lim_x→a f(x)
• Constant Multiple Rule of Integration: ∫k f(x) dx = k ∫f(x) dx

When to Use Constant Multiple Rule?

We can use the constant multiple rule when an operation is applied on a function multiplied by a constant. For example, to find the derivative of a function multiplied by a constant we use the formula, d(k f(x))/dx = k d(f((x))/dx.

How to Use the Constant Multiple Rule?

We can use the constant multiple rule by applying the formulas to find the derivatives, integrals, and limits.

• Constant Multiple Rule of Differentiation: d(k f(x))/dx = k d(f((x))/dx
• Constant Multiple Rule of Limits: lim_x→a k f(x) = k lim_x→a f(x)
• Constant Multiple Rule of Integration: ∫k f(x) dx = k ∫f(x) dx

What is the Constant Multiple Rule for Derivatives?

The constant multiple rule for derivatives states that the derivative of the product of a constant with a function f(x) is equal to the product of the constant with the derivative of the function f(x). To apply this rule, we can use the formula d(k f(x))/dx = k d(f((x))/dx.

How Do You Prove a Constant Multiple Rule for Derivatives?

We can prove the constant multiple rule for derivatives by using the first principle of differentiation. We can use the following formulas to prove the constant multiple rule for derivatives:

• f'(x) = lim_h→0 [f(x+h) - f(x)] / h --- (1)
• Product Rule of Limits: lim_x→a [ f(x) × g(x) ] = lim_x→a f(x) × lim_x→a g(x)
• Limit of a constant: lim_x→a k = k