Differentiating from first principles.

Find the derivative of \( f(x) = x^3 + 3x \) at x = 1 using the limit definition of the derivative.

Substitute the expression into the definition.

\[ \begin{align} f\ '(1) &= \lim_{h \to 0} \frac{f(1+h)- f(1)}{h} \\ &= \lim_{h \to 0} \frac{(1+h)^3 + 3(1+h) - (1+3)}{h} \\ &= \lim_{h \to 0} \frac{1 + 3h + 3h^2 + h^3 + 3+ 3h - 1 - 3}{h} \\ &= \lim_{h \to 0} \frac{6h + 3h^2 + h^3}{h} \\ &= \lim_{h \to 0} (6 + 3h + h^2) \\ \end{align} \ \]

Apply the definition of the limit.

As h approaches zero, the value of the expression approaches 6. \[ \therefore f\ '(1)= 6. \quad \quad \]

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 \( f(x) = 2x^2 + 3x \) at x = 1 using the limit definition of the derivative.