Single Variable Calculus, Chapter 3, 3.2, Section 3.2, Problem 27

Find the derivative of $\displaystyle f(x) = x^4$ using the definition and the domain of its derivative.

Using the definition of derivative


$
\begin{equation}
\begin{aligned}

\qquad f'(x) &= \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
&&
\\
\\
\qquad f'(x) &= \lim_{h \to 0} \frac{(x + h)^4 - x^4}{h}
&& \text{Substitute $f(x + h)$ and $f(x)$}
\\
\\
\qquad f'(x) &= \lim_{h \to 0} \frac{\cancel{x^4} + 4x^3 + 6x^2 h^2 + 4xh^3 + h^4 -\cancel{x^4}}{h}
&& \text{Expand the equation and combine like terms}
\\
\\
\qquad f'(x) &= \lim_{h \to 0} \frac{4x^3h + 6x^2 h^2 + 4xh^3 + h^4}{h}
&& \text{Factor the numerator}
\\
\\
\qquad f'(x) &= \lim_{h \to 0} \frac{\cancel{h} (4x^3 + 6x^2h + 4xh^2 + h^3)}{\cancel{h}}
&& \text{Cancel out like terms}
\\
\\
\qquad f'(x) &= \lim_{h \to 0} (4x^3 + 6x^2 h + 4xh^2 +h ) = 4x^3 + 6x^2(0) + 4x(0)^2 + (0)^3
&& \text{Evaluate the limit}

\end{aligned}
\end{equation}
$


$\qquad \fbox{$f'(x) = 4x^3$}$