Answer:
-12x²
General Formulas and Concepts:
Pre-Algebra
Algebra I
- Function Notation
- Combining Like Terms
- Factoring
- Expanding by FOIL (First Outside Inside Last)
Calculus
- Evaluating Limits
- Definition of a Derivative: [tex]f'(x)= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}[/tex]
Step-by-step explanation:
Step 1: Define
[tex]f(x) = 1 - 4x^3[/tex]
Step 2: Differentiate
- Substitute [DOD]: [tex]f'(x)= \lim_{h \to 0} \frac{[1-4(x+h)^3]-(1-4x^3)}{h}[/tex]
- Expand: [tex]f'(x)= \lim_{h \to 0} \frac{[1-4(x^3+3hx^2+3h^2x+h^3)]-(1-4x^3)}{h}[/tex]
- Distribute: [tex]f'(x)= \lim_{h \to 0} \frac{1-4x^3-12hx^2-12h^2x-4h^3-1+4x^3}{h}[/tex]
- Combine like terms: [tex]f'(x)= \lim_{h \to 0} \frac{-12h^2x-4h^3-12hx^2}{h}[/tex]
- Factor: [tex]f'(x)= \lim_{h \to 0} \frac{h(-12hx-4h^2-12x^2)}{h}[/tex]
- Divide: [tex]f'(x)= \lim_{h \to 0} -12hx-4h^2-12x^2[/tex]
- Evaluate: [tex]f'(x)= \-12x^2[/tex]
