# Constant Multiple Rule for Derivatives (With Proof and Examples)

*Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.*

## 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) 3x^{4} 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) 3x^{4} = 3 (d/dx) x^{4}

Apply the Power Rule in differentiating the power function.

(d/dx) 3x^{4} = 3 (d/dx) 4x^{4-1}

(d/dx) 3x^{4} = 3 (d/dx) 4x^{3}

Simplify the algebraic expression.

(d/dx) 3x^{4} = 3 (d/dx) 4x^{3}

(d/dx) 3x^{4} = 3(4x^{3})

(d/dx) 3x^{4} = 12x^{3}

**Answer**

The derivative of the function (d/dx) 3x^{4} is 12x^{3}.

## 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) x^{3}?

**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) x^{3} = (^{3}√8) (d/dx) x^{3}

Recall the Power Rule and solve for the derivative of the power function x^{3}.

(d/dx) (^{3}√8) x^{3} = ^{3}√8 (3x^{2})

(d/dx) (^{3}√8) x^{3} = 2(3x^{2})

(d/dx) (^{3}√8) x^{3} = 6x^{2}

**Answer**

The derivative of the function y = (^{3}√8) x^{3} is 6x^{2}.

## Example 6: Derivative of a Function to the Seventh Power

Find the derivative of the function y = 3x^{7} 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 x^{7} using the Power Rule.

(d/dx) 3x^{7 }= 3 (d/dx) x^{7}

** **

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

(d/dx) 3x^{7 }= 3 (d/dx) x^{7} = 3(7x^{7-1})

(d/dx) 3x^{7 }= 3(7x^{6})

(d/dx) 3x^{7} = 21x^{6}

**Answer**

The derivative of the function y = 3x^{7} is 21x^{6}.

## 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) x^{0}

(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) t^{0}

(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) = 4x^{4}.

**Solution**

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

(d/dx) 4x^{4}= 4 (d/dx) x^{4}

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

(d/dx) 4x^{4}= 4 (d/dx) x^{4}

(d/dx) 4x^{4}= 4 (d/dx) 4x^{4-1}

(d/dx) 4x^{4}= 4 (d/dx) 4x^{3}

(d/dx) 4x^{4}= 16x^{3}

**Answer**

The derivative of the function 4x^{4} is 16x^{3}.

## Example 10: Derivative of a Sum of Power Functions

Find the derivative of the function f(x) = 6x^{3} + 9x^{2} + 2x + 8.

**Solution**

The given equation is a run of power functions. To solve, differentiate the terms individually. Start with the 6x^{3} and apply the Constant Multiple Rule.

(d/dx) 6x^{3}= 6 (d/dx) x^{3}

(d/dx) 6x^{3}= 6 (3x^{3-1})

(d/dx) 6x^{3}= 6 (3x^{2})

(d/dx) 6x^{3}= 18x^{2}

(d/dx) 9x^{2}= 9 (d/dx) x^{2}

(d/dx) 9x^{2}= 9 (2x^{2-1})

(d/dx) 9x^{2}= 9 (2x^{1})

(d/dx) 9x^{2}= 18x

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

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

(d/dx) 2x = 2 (x^{0})

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

(d/dx) 2x = 2

**Answer**

The first derivative of the function y = 6x^{3} + 9x^{2} + 2x + 8 is y’ = 18x^{2} +18x + 2.

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**© 2021 Ray**