2. Intro to Derivatives

Derivatives as Functions

2. Intro to Derivatives

Derivatives as Functions

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Multiple Choice

Use the definition of a derivative, to find the derivative of the function $g(x)=x^3$ at $x=-1$ .

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Verified step by step guidance

Start with the definition of the derivative: $ g'(x) = \lim_{h \to 0} \frac{g(x+h) - g(x)}{h} $.

Substitute $ g(x) = x^3 $ into the definition: $ g'(x) = \lim_{h \to 0} \frac{(x+h)^3 - x^3}{h} $.

Expand $ (x+h)^3 $ using the binomial theorem: $ (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 $. Substitute this back into the equation: $ g'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h} $.

Simplify the numerator by canceling $ x^3 $: $ g'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h} $. Factor $ h $ out of the numerator: $ g'(x) = \lim_{h \to 0} \frac{h(3x^2 + 3xh + h^2)}{h} $. Cancel $ h $ (for $ h \neq 0 $): $ g'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2) $.

Evaluate the limit as $ h \to 0 $: $ g'(x) = 3x^2 $. To find the derivative at $ x = -1 $, substitute $ x = -1 $ into $ g'(x) = 3x^2 $: $ g'(-1) = 3(-1)^2 $.