Constant Multiple Rule is a fundamental concept in calculus used to simplify the process of differentiation, integration, and finding limits when dealing with functions multiplied by a constant. This rule essentially states that when a constant is multiplied by a function, the operations of differentiation, integration, or limits can be performed as if the constant were factored out.

In this article, we will discuss the constant multiple rule for derivatives, integration as well as limits. We will also derive the rule for derivatives and discuss some of the solved examples as well.

What is Constant Multiple Rule?

The Constant Multiple Rule holds that the constant multiplied by the derivative of a function yields the constant multiplied by the derivative of the function. Mathematically, should c be a constant and f(x) be a differentiable function, then:

d/dx [c⋅f(x)]=c⋅ d/dx [f(x)]

This criterion is essential as, particularly in cases of functions multiplied by constants, it simplifies the differentiation process. Appreciating and knowing this rule will save time and simplify calculus issues.

Formula for Constant Multiple Rule

The formula for the Constant Multiple Rule can be expressed as:

d/dx [c⋅f(x)] = c⋅ d/dx [f(x)]

Where c is a constant and f(x) is a differentiable function.

Derivation of the Constant Multiple Rule

Examining the definition of the derivative helps one to develop the Constant Multiple Rule:

d/dx [c⋅f(x)] = lim_Δ→0 (c.f(x+Δx)-c.f(x))/Δx

Since c is a constant, it can be factored out of the limit:

d/dx [c⋅f(x)] = c.lim_Δ→0 (f(x + Δx) – f(x))/Δx

Thus: d/dx [c⋅f(x)] = c. d/dx [f(x)]

This derivation proves that constant multiplied by a function’s derivative equals the constant multiplied by the derivative of the function.

Some Other Constant Multiple Rules

There are various different rules related to constant multiple

• Constant Multiple Rule for Limits
• Constant Multiple Rule for Integration

Constant Multiple Rule for Limits

The Constant Multiple Rule also applies to limits. If c is constant and lim_{x → a} f(x) exists, then:

lim_x→a [c. f(x)] = c. lim_x→a f(x)

Constant Multiple Rule for Integration

For integration, the rule indicates that the integral of a constant times a function is the constant times the integral of the function. If g(x) = c · f(x), then

[Tex]\int g(x) \, dx = c \int f(x) \, dx[/Tex]

Conclusion

A key instrument in calculus, the Constant Multiple Rule helps to simplify differentiation, limits, and integration with constants. Knowing and using this rule will help to greatly improve accuracy and efficiency in mathematical problem solving.

Read More,

• Calculus
• Differential Calculus
• Integral Calculus
• List of All Symbols in Calculus
• Derivatives
• Derivative Test
• Derivative of a Function in Parametric Form

Examples of Constant Multiple Rule

Example 1: Find derivative of f(x) = 7sin(x).

Solution:

Here the constant is 7 and function is sin(x)

Derivative of sin(x) is cos(x)

Therefore d/dx [7 sin(x)] = 7cos(x)

Example 2: Find derivative of g(x) = -3e^x

Solution:

Here the constant is -3 and the function is e^x

Derivative of e^x is e^x

Therefore d/dx [-3e^x] = -3e^x

Example 3: Find the derivative of g(x) = 4x³ Using Constant Multiple Rule.

Solution:

d/dx[4x³] = 4⋅d/dx[x³]

The derivative of x³ is 3x², so: d/dx[4x³] = 4⋅3x² = 12 x²

Example 4: Find lim⁡_x→2f(x), where f(x)=2x²+1.

Solution:

[Tex]\lim_{x \to 2} f(x) = \lim_{x \to 2} (2x^2 + 1)[/Tex]

[Tex]\Rightarrow \lim_{x \to 2} f(x) = 2\lim_{x \to 2}x^2 + \lim_{x \to 2} 1 [/Tex]

[Tex]\Rightarrow \lim_{x \to 2} f(x) = 2\cdot 2^2 + 1 [/Tex]

[Tex]\Rightarrow \lim_{x \to 2} f(x) = 2 \cdot 4 + 1 = 8 + 1 = 9[/Tex]

Practice Problems on Constant Multiple Rule

Problem 1: Find the derivative of h(x) = 7x²

Problem 2: Differentiate f(x) = -4cos(x)

Problem 3: Find the derivative of g(x) = 3 ln(x)

Problem 4: Differentiate h(x) = 7e^-x

Problem 5: Find the derivative of f(x) = 5 tan(x)

FAQs on Constant Multiple Rule:

What is the Constant Multiple Rule ?

The Constant Multiple Rule says you may just multiply the constant by the derivative of a function to differentiate a constant multiplied by a function.

Why may the Constant Multiple Rule be helpful?

It simplifies and accelerates the differentiation process, hence facilitating handling of functions with constants.

Does the Constant Multiple Rule apply to all functions?

Indeed, the rule relates to all differentiable functions.

Can the Constant Multiple Rule be used for integration?

Similar ideas hold true for integration, sometimes referred to as the Constant Multiple Rule for Integration, which lets constants be factored out of the integral.

How might one apply the Constant Multiple Rule?

Applying the Constant Multiple Rule for the function 4x² yields 4⋅2x = 8x.

In what ways may the Constant Multiple Rule interact with other differentiation rules?

To simplify difficult differentiation problems, it is frequently used in concert with other guidelines including the Power Rule, Product Rule, and Chain Rule.

Are there any exceptions to the Constant Multiple Rule?

It holds generally of all differentiable functions and constants; there are no exceptions.