Answer:
[tex]\displaystyle f'(x) = -12x^2 + 4x[/tex]
General Formulas and Concepts:
Algebra I
Terms/Coefficients
Calculus
Limits
- Limit Rule [Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to c} x = c[/tex]
Differentiation
- Derivatives
- Derivative Notation
The definition of a derivative is the slope of the tangent line: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}[/tex]
Step-by-step explanation:
Step 1: Define
Identify.
[tex]\displaystyle f(x) = -4x^3 + 2x^2[/tex]
Step 2: Differentiate
- [Function] Substitute in x: [tex]\displaystyle f(x + h) = -4(x + h)^3 + 2(x + h)^2[/tex]
- Substitute in functions [Definition of a Derivative]: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{-4(x + h)^3 + 2(x + h)^2 - \big( -4x^3 + 2x^2 \big)}{h}[/tex]
- Simplify: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{-12hx^2 - 12h^2x + 4hx - 4h^3 + 2h^2}{h}[/tex]
- Factor: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{h \big( -12x^2 - 12hx + 4x - 4h^2 + 2h \big)}{h}[/tex]
- Simplify: [tex]\displaystyle f'(x) = \lim_{h \to 0} -12x^2 - 12hx + 4x - 4h^2 + 2h[/tex]
- Evaluate limit [Limit Rule - Variable Direct Substitution]: [tex]\displaystyle f'(x) = -12x^2 - 12(0)x + 4x - 4(0)^2 + 2(0)[/tex]
- Simplify: [tex]\displaystyle f'(x) = -12x^2 + 4x[/tex]
∴ the derivative of the given function will be equal to -12x² + 4x.
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Learn more about derivatives: https://brainly.com/question/25804880
Learn more about calculus: https://brainly.com/question/23558817
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Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
