23. Intro to Derivatives & Area Under the Curve

Tangent Lines & Derivatives

23. Intro to Derivatives & Area Under the Curve

Tangent Lines & Derivatives

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

Substitute the function $ g(x) = x^3 $ into the derivative 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 the expanded form back into the limit expression: $ g'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h} $.

Simplify the expression by canceling $ x^3 $ and dividing each term by $ h $: $ g'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2) $. Evaluate the limit as $ h \to 0 $ to find $ g'(x) $.

5:13m

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Struggling with Precalculus?

Use the definition of a derivative, to find the derivative of the function g(x)=x3g(x)=x^3g(x)=x3 at x=−1x=-1x=−1.

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Tangent Lines & Derivatives practice set

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