Constant Multiple Rule for Derivatives (With Proof and Examples)

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What Is the Constant Multiple Rule?

The derivative of constant times a function is the constant times the derivative of the function. If c is a constant and f is a differentiable function, then,

(d/dx) [c f(x)] = c (d/dx) [f(x)]

Constant Multiple Rule Proof

When new functions are formed from old functions by multiplication by a constant or any other operations, their derivatives can be calculated using derivatives of the old functions. In particular, the Constant Multiple Rule states that the derivative of a constant multiplied by a function is the constant multiplied by the function's derivative.

Here is ample proof of Constant Multiple Rule using limits. Let g(x) = c f(x)

g’(x) = limh→o [g(x+h)-g(x)] / h

g’(x) = limh→o [cf(x+h)-cf(x)] / h

g’(x) = limh→o c [(f(x+h)-f(x)) / h]

g’(x) = c limh→o [(f(x+h)-f(x)) / h]

g’(x) = cf’(x)

Below is the geometric interpretation of the Constant Multiple Rule. Multiplying by c=2 stretches the graph vertically by a factor of 2. All the rises have been doubled, but the runs stay the same. So the slopes are doubled, too.

Example 1: Derivative of a Function to the Fourth Power

Find the derivative of the function (d/dx) 3x⁴ using the Constant Multiple Rule.

Solution

Apply the Constant Multiple Rule by taking the derivative of the power function first and then multiply with the coefficient 3.

(d/dx) 3x⁴ = 3 (d/dx) x⁴

Apply the Power Rule in differentiating the power function.

(d/dx) 3x⁴ = 3 (d/dx) 4x^4-1

(d/dx) 3x⁴ = 3 (d/dx) 4x³

Simplify the algebraic expression.

(d/dx) 3x⁴ = 3 (d/dx) 4x³

(d/dx) 3x⁴ = 3(4x³)

(d/dx) 3x⁴ = 12x³

Answer

The derivative of the function (d/dx) 3x⁴ is 12x³.

Example 2: Derivative of a Negative Function

What is the derivative of the function (d/dx) -x?

Solution

Apply the Constant Multiple Rule by taking the derivative of the power function first and then multiply with the coefficient -1.

(d/dx) -x = (d/dx) [(-1) x]

Apply the Power Rule in differentiating the power function.

(d/dx) -x = (-1) (d/dx) x

Recall that the derivative of x is 1. Simplify further the algebraic expression.

(d/dx) -x = -1(1)

(d/dx) -x = -1

Answer

The derivative of the function -x is -1.

Example 3: Derivative of a Fraction

What is the derivative of the function y= (1/2) x?

Solution

Apply the Constant Multiple Rule by taking the derivative of the power function first and then multiply with the coefficient \frac {1} {2}. Apply the Power Rule in differentiating the power function.

(d/dx) (1/2)x = (1/2) (d/dx) x

Recall that the derivative of x is 1. Simplify further the algebraic expression.

(d/dx) (1/2) x = (1/2) (1)

(d/dx) (1/2) x = ½

Answer

The derivative of the function y = (1/2) x is ½.

Example 4: Derivative of a Function of X

Find the derivative of the constant multiple function f(x)=6x.

Solution

Apply the Constant Multiple Rule by taking the derivative of the power function first and then multiply with the coefficient 6. Apply the Power Rule in differentiating the power function.

(d/dx) 6x = 6 (d/dx) x

Recall that the derivative of x is 1. Simplify further the algebraic expression.

(d/dx) 6x = 6(1)

(d/dx) 6x = 6

Answer

The derivative of the function f(x)=6x is 6.

Example 5: Derivative of a Complex Cube Root Function

What is the derivative of the constant multiple function y = (³√8) x³?

Solution

Apply the Constant Multiple Rule by taking the derivative of the power function first and then multiply with the coefficient ³√8. Apply the Power Rule in differentiating the power function.

(d/dx) (³√8) x³ = (³√8) (d/dx) x³

Recall the Power Rule and solve for the derivative of the power function x³.

(d/dx) (³√8) x³ = ³√8 (3x²)

(d/dx) (³√8) x³ = 2(3x²)

(d/dx) (³√8) x³ = 6x²

Answer

The derivative of the function y = (³√8) x³ is 6x².

Example 6: Derivative of a Function to the Seventh Power

Find the derivative of the function y = 3x⁷ using the Constant Multiple Rule.

Solution

First, separate the constant value of 3 from the whole function. Then, take the derivative of the power function x⁷ using the Power Rule.

(d/dx) 3x⁷ = 3 (d/dx) x⁷

Apply the Power Rule then multiply the result to the constant 3, as stated from the Constant Multiple Rule.

(d/dx) 3x⁷ = 3 (d/dx) x⁷ = 3(7x^7-1)

(d/dx) 3x⁷ = 3(7x⁶)

(d/dx) 3x⁷ = 21x⁶

Answer

The derivative of the function y = 3x⁷ is 21x⁶.

Example 7: Derivative of a Negative Function

What is the derivative of the constant multiple function (d/dx) (-5x)?

Solution

From the given function, segregate the negative value constant from the x variable.

(d/dx) -5x= -5 (d/dx) x

Apply the Power Rule, then multiply the outcome with a negative five.

(d/dx) -5x= -5 (d/dx) x

(d/dx) -5x= -5 (d/dx) x^1-1

(d/dx) -5x= -5 (d/dx) x⁰

(d/dx) -5x= -5 (d/dx) (1)

(d/dx) -5x= -5

Answer

The derivative of the function y = -5x is -5.

Example 8: Derivative of a Pi Function

Find the derivative of the function y = 3πt.

Solution

The given function is a radian function of variable t. Recall that pi is a constant value of 3.14. It means that the part with 3π will be the constant of the pi function. Separate the constant value 3π from the variable t and differentiate t alone.

(d/dt) 3πt= 3π (d/dt) t

Apply the Power Rule and the Constant Multiple Rule to the equation.

(d/dt) 3πt= 3π (d/dt) t^1-1

(d/dt) 3πt= 3π (d/dt) t⁰

(d/dt) 3πt= 3π

Answer

The derivative of the function y = 3πt is 3π.

Example 9: Derivative of a Function to the Fourth Power

Solve for the derivative of the function f(x) = 4x⁴.

Solution

In getting the derivative of the function 4x⁴, the Constant Multiple Rule applies. Like the usual method, separate the constant four from x⁴ and then solve the power function's derivative.

(d/dx) 4x⁴= 4 (d/dx) x⁴

Apply the Power Rule and the Constant Multiple Rule to the equation.

(d/dx) 4x⁴= 4 (d/dx) x⁴

(d/dx) 4x⁴= 4 (d/dx) 4x^4-1

(d/dx) 4x⁴= 4 (d/dx) 4x³

(d/dx) 4x⁴= 16x³

Answer

The derivative of the function 4x⁴ is 16x³.

Example 10: Derivative of a Sum of Power Functions

Find the derivative of the function f(x) = 6x³ + 9x² + 2x + 8.

Solution

The given equation is a run of power functions. To solve, differentiate the terms individually. Start with the 6x³ and apply the Constant Multiple Rule.

(d/dx) 6x³= 6 (d/dx) x³

(d/dx) 6x³= 6 (3x^3-1)

(d/dx) 6x³= 6 (3x²)

(d/dx) 6x³= 18x²

(d/dx) 9x²= 9 (d/dx) x²

(d/dx) 9x²= 9 (2x^2-1)

(d/dx) 9x²= 9 (2x¹)

(d/dx) 9x²= 18x

(d/dx) 2x= 2 (d/dx) x

(d/dx) 2x = 2 (x^1-1)

(d/dx) 2x = 2 (x⁰)

(d/dx) 2x = 2 (1)

(d/dx) 2x = 2

Answer

The first derivative of the function y = 6x³ + 9x² + 2x + 8 is y’ = 18x² +18x + 2.

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