Basic Differentiation
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Example 1:
$$ y=x^4+x^2,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y&=x^4+x^2\\
\frac{dy}{dx}&=\frac{d}{dx}(x^4+x^2)\\
\frac{dy}{dx}&=\frac{d}{dx}x^4+\frac{d}{dx}x^2\\\\
\because\frac{d}{dx}x^n&=nx^{n-1}\\\\
\therefore\;\; \frac{dy}{dx}&=4x^{4-1}+2x^{2-1}\\
\frac{dy}{dx}&=4x^3+2x\\
\end{align*}
Example 2:
$$ y=x^3+\sqrt{x},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y&=x^3+\sqrt{x}\\
\frac{dy}{dx}&=\frac{d}{dx}(x^3+\sqrt{x})\\
\frac{dy}{dx}&=\frac{d}{dx}x^3+\frac{d}{dx}\sqrt{x}\\
\frac{dy}{dx}&=\frac{d}{dx}x^3+\frac{d}{dx}x^{\frac{1}{2}}\\\\
\because\frac{d}{dx}x^n&=nx^{n-1}\\\\
\therefore\;\; \frac{dy}{dx}&=3x^{3-1}+\frac{1}{2}x^{\frac{1}{2}-1}\\
\frac{dy}{dx}&=3x^2+\frac{1}{2}x^{-\frac{1}{2}}\\
\frac{dy}{dx}&=3x^2+\frac{1}{2x^{\frac{1}{2}}}\\
\frac{dy}{dx}&=3x^2+\frac{1}{2\sqrt{x}}\\
\end{align*}
Example 3:
$$ y=5x^6+2x^4,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y&=5x^6+2x^4\\
\frac{dy}{dx}&=\frac{d}{dx}(5x^6+2x^4)\\
\frac{dy}{dx}&=\frac{d}{dx}5x^6+\frac{d}{dx}2x^4\\
\frac{dy}{dx}&=5\frac{d}{dx}x^6+2\frac{d}{dx}x^4\\\\
\because\frac{d}{dx}x^n&=nx^{n-1}\\\\
\therefore\;\; \frac{dy}{dx}&=5(6x^{6-1})+2(4x^{4-1})\\
\frac{dy}{dx}&=30x^5+8x^3\\
\end{align*}