Differentiating from first principles.

Find the derivative of \( f(x) = x^3 \) using the limit definition of the derivative.

Substitute the expression into the definition.

\[ \begin{align} f\ '(x) &= \lim_{h \to 0} \frac{f(x+h)- f(x)}{h} \\ &= \lim_{h \to 0} \frac{(x+h)^3- x^3}{h} \\ &= \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h} \\ &= \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h} \end{align} \ \]

Cancel h and apply the definition of the limit.

\[ \begin{align} &= \lim_{h \to 0} (3x^2 + 3xh + h^2) \\ &= 3x^2 \end{align} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

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Now You Try:

Find the derivative of \( f(x) = x^3 - 2x \) using the limit definition of the derivative.