Sure, let's break down the differentiation process to understand the given problem better.
We are given the function [tex]\( f(x) = -4x + 3 \)[/tex] and asked to find its derivative, denoted [tex]\( f'(x) \)[/tex].
Step 1: Write down the function
[tex]\[ f(x) = -4x + 3 \][/tex]
Step 2: Recall the basic derivative rules
- The derivative of a constant [tex]\( c \)[/tex] is 0.
- The derivative of [tex]\( x \)[/tex] is 1.
- The derivative of a constant times a function [tex]\( c \cdot f(x) \)[/tex] is the constant times the derivative of the function [tex]\( c \cdot f'(x) \)[/tex].
Step 3: Apply the derivative rules
- The term [tex]\(-4x\)[/tex]:
The derivative of [tex]\( x \)[/tex] is 1, so the derivative of [tex]\(-4x\)[/tex] becomes:
[tex]\[
\frac{d}{dx}(-4x) = -4 \cdot \frac{d}{dx}(x) = -4 \cdot 1 = -4
\][/tex]
- The term [tex]\(3\)[/tex]:
The derivative of a constant is 0, so:
[tex]\[
\frac{d}{dx}(3) = 0
\][/tex]
Step 4: Combine the results
[tex]\[
f'(x) = \frac{d}{dx}(-4x + 3) = \frac{d}{dx}(-4x) + \frac{d}{dx}(3) = -4 + 0 = -4
\][/tex]
Therefore, the derivative of the function [tex]\( f(x) = -4x + 3 \)[/tex] is:
[tex]\[ f'(x) = -4 \][/tex]
