Derivative Function Explained

Understanding how to find the derivative of a function is a fundamental concept in calculus. The derivative represents the instantaneous rate of change of a function. For polynomial functions, like the given \(f(x) = -3x^2 + 6x - 4\), we primarily use a few basic differentiation rules.

Function Differentiation Rules

To find the derivative of the function \(f(x) = -3x^2 + 6x - 4\), we will apply the following essential rules of differentiation:

• Power Rule: If \(g(x) = x^n\), where \(n\) is any real number, then its derivative is given by \(g'(x) = nx^{n-1}\).
• Constant Multiple Rule: If \(h(x) = c \cdot g(x)\), where \(c\) is a constant and \(g(x)\) is a differentiable function, then its derivative is \(h'(x) = c \cdot g'(x)\).
• Sum/Difference Rule: The derivative of a sum or difference of functions is the sum or difference of their individual derivatives. That is, if \(f(x) = u(x) \pm v(x)\), then \(f'(x) = u'(x) \pm v'(x)\).
• Derivative of a Constant: If \(k(x) = c\), where \(c\) is a constant, then its derivative is always \(k'(x) = 0\).

Step-by-Step Derivative Calculation

Let's apply these rules to each term of the function \(f(x) = -3x^2 + 6x - 4\) to find its derivative, denoted as \(f'(x)\) or \(\frac{df}{dx}\).

The original function is:

To find the derivative \(f'(x)\), we differentiate each term with respect to \(x\):

Term 1: Derivative of \(-3x^2\)

For this term, we use the Constant Multiple Rule and the Power Rule:

• Apply the Constant Multiple Rule: \(-3 \cdot \frac{d}{dx}(x^2)\)
• Apply the Power Rule to \(\frac{d}{dx}(x^2)\): Here, \(n=2\), so \(2x^{2-1} = 2x^1 = 2x\).
• Combine them: \(\frac{d}{dx}(-3x^2) = -3 \cdot (2x) = -6x\).

Term 2: Derivative of \(6x\)

For this term, we also use the Constant Multiple Rule and the Power Rule:

• Apply the Constant Multiple Rule: \(6 \cdot \frac{d}{dx}(x)\)
• Apply the Power Rule to \(\frac{d}{dx}(x)\): Here, \(x\) can be written as \(x^1\), so \(n=1\). The derivative is \(1x^{1-1} = 1x^0\). Since any non-zero number raised to the power of 0 is 1, \(x^0 = 1\).
• Combine them: \(\frac{d}{dx}(6x) = 6 \cdot (1) = 6\).

Term 3: Derivative of \(-4\)

For this term, we use the Derivative of a Constant Rule:

• The derivative of any constant number (like \(-4\)) is \(0\).
• So, \(\frac{d}{dx}(-4) = 0\).

Combining the Derivatives

Now, we sum the derivatives of each individual term to get the total derivative \(f'(x)\):

Derivative Result Comparison

The calculated derivative of the function \(f(x) = -3x^2 + 6x - 4\) is \(-6x + 6\). Let's compare this result with the provided options:

• Option 1: \(6x + 6\)
• Option 2: \(6x - 6\)
• Option 3: \(-6x - 6\)
• Option 4: \(-6x + 6\)

The computed derivative \(-6x + 6\) perfectly matches Option 4.