What is the derivative of the function f(x) = 4x^3?

What is the derivative of the function f(x) = 4x^3?

• A. 12x^2
• B. 6x^2
• C. 8x^2
• D. 3x^2

Answer: (A)

To find the derivative of the function \( f(x) = 4x^3 \), we can apply the power rule of differentiation. The power rule states that if you have a function in the form of \( ax^n \), its derivative is given by multiplying the coefficient \( a \) by the exponent \( n \) and then reducing the exponent by 1. In this case, \( a = 4 \) and \( n = 3 \). Applying the power rule: 1. Multiply the coefficient (4) by the exponent (3): \( 4 \times 3 = 12 \). 2. Decrease the exponent by 1: \( 3 - 1 = 2 \). Putting these together, we find that the derivative \( f'(x) \) is \( 12x^{2} \). This matches the correct answer that was identified. Thus, the derivative of the function \( 4x^3 \) is indeed \( 12x^2 \). Understanding this process is essential for solving similar problems and finding derivatives of polynomial functions using the power rule.

To find the derivative of the function ( f(x) = 4x^3 ), we can apply the power rule of differentiation. The power rule states that if you have a function in the form of ( ax^n ), its derivative is given by multiplying the coefficient ( a ) by the exponent ( n ) and then reducing the exponent by 1.

In this case, ( a = 4 ) and ( n = 3 ). Applying the power rule:

1. Multiply the coefficient (4) by the exponent (3):

( 4 \times 3 = 12 ).

2. Decrease the exponent by 1:

( 3 - 1 = 2 ).

Putting these together, we find that the derivative ( f'(x) ) is ( 12x^{2} ). This matches the correct answer that was identified.

Thus, the derivative of the function ( 4x^3 ) is indeed ( 12x^2 ). Understanding this process is essential for solving similar problems and finding derivatives of polynomial functions using the power rule.