Basic Differentiation
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Example 1:
$$ y=x^4,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y&=x^4\\
\frac{dy}{dx} &=\frac{d}{dx} x^4\\\\
\because\;\; \frac{d}{dx} x^n &=nx^{n-1}\\\\
\therefore\;\; \frac{dy}{dx} &=4x^{4-1}\\
\frac{dy}{dx} &=4x^3\\
\end{align*}
Example 2:
$$ y=3x^5,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y &=3x^5\\
\frac{dy}{dx} &=\frac{d}{dx} 3x^5\\
\frac{dy}{dx} &=3\frac{d}{dx}x^5\\\\
\because\;\; \frac{d}{dx} x^n &=nx^{n-1}\\\\
\therefore\;\; \frac{dy}{dx} &=3(5x^{5-1})\\
\frac{dy}{dx} &=15x^4\\
\end{align*}
Example 3:
$$ y=x^3+8,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y &=x^3+8\\
\frac{dy}{dx} &=\frac{d}{dx}(x^3+8)\\
\frac{dy}{dx} &=\frac{d}{dx}x^3+\frac{d}{dx}8\\\\
\because\;\; \frac{d}{dx} x^n &=nx^{n-1}\;\; \textup{and} \;\:\frac{d}{dx} c=0\\\\
\therefore\;\; \frac{dy}{dx} &=3x^{3-1}+0\\
\frac{dy}{dx} &=3x^2\\
\end{align*}
Example 4:
$$ y=2x^2+5x,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y&=2x^2+5x\\
\frac{dy}{dx}&=\frac{d}{dx}(2x^2+5x)\\
\frac{dy}{dx}&=\frac{d}{dx}2x^2+\frac{d}{dx}5x\\
\frac{dy}{dx}&=2\frac{d}{dx}x^2+5\frac{d}{dx}x\\\\
\because\;\; \frac{d}{dx}x^n&=nx^{n-1}\;\; \textup{and} \;\:\frac{d}{dx}x=1\\\\
\therefore\;\; \frac{dy}{dx}&=2(2x^{2-1})+5(1)\\
\frac{dy}{dx}&=4x+5\\
\end{align*}
Example 5:
$$ y=7x^6-9x^4+12x^2,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y&=7x^6-9x^4+12x^2\\
\frac{dy}{dx}&=\frac{d}{dx}(7x^6-9x^4+12x^2)\\
\frac{dy}{dx}&=\frac{d}{dx}7x^6-\frac{d}{dx}9x^4+\frac{d}{dx}12x^2\\
\frac{dy}{dx}&=7\frac{d}{dx}x^6-9\frac{d}{dx}x^4+12\frac{d}{dx}x^2\\\\
\because\;\; \frac{d}{dx}x^n&=nx^{n-1}\\\\
\therefore\;\; \frac{dy}{dx}&=7(6x^{6-1})-9(4x^{4-1})+12(2x^{2-1})\\
\frac{dy}{dx}&=42x^5-36x^3+24x\\
\end{align*}
Example 6:
$$ y=3x^3-4x^2+7x-10,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y&=3x^3-4x^2+7x-10\\
\frac{dy}{dx}&=\frac{d}{dx}(3x^3-4x^2+7x-10)\\
\frac{dy}{dx}&=\frac{d}{dx}3x^3-\frac{d}{dx}4x^2+\frac{d}{dx}7x-\frac{d}{dx}10\\
\frac{dy}{dx}&=3\frac{d}{dx}x^3-4\frac{d}{dx}x^2+7\frac{d}{dx}x-\frac{d}{dx}10\\\\
\because\;\; \frac{d}{dx}x^n&=nx^{n-1}\,,\,\frac{d}{dx}x=1\,\textup{and}\,\frac{d}{dx}c=0\\\\
\therefore\;\; \frac{dy}{dx}&=3(3x^{3-1})-4(2x^{2-1})+7(1)-0\\
\frac{dy}{dx}&=9x^2-8x+7\\
\end{align*}
Example 7:
$$ y=\frac{1}{x},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y&=\frac{1}{x}\\
\frac{dy}{dx}&=\frac{d}{dx}\left(\frac{1}{x}\right)\\
\frac{dy}{dx}&=\frac{d}{dx}x^{-1}\\\\
\because \frac{d}{dx}x^n&=nx^{n-1}\\\\
\therefore \frac{dy}{dx}&=-1x^{-1-1}\\
\frac{dy}{dx}&=-1x^{-2}\\
\frac{dy}{dx}&=\frac{-1}{x^2}\\
\frac{dy}{dx}&=-\frac{1}{x^2}\\
\end{align*}
Example 8:
$$ y=\frac{1}{x^2},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y&=\frac{1}{x^2}\\
\frac{dy}{dx}&=\frac{d}{dx}\left(\frac{1}{x^2}\right)\\
\frac{dy}{dx}&=\frac{d}{dx}x^{-2}\\\\
\because \frac{d}{dx}x^n&=nx^{n-1}\\\\
\therefore \frac{dy}{dx}&=-2x^{-2-1}\\
\frac{dy}{dx}&=-2x^{-3}\\
\frac{dy}{dx}&=\frac{-2}{x^3}\\
\frac{dy}{dx}&=-\frac{2}{x^3}\\
\end{align*}
Example 9:
$$ y=\frac{4}{x^3},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y&=\frac{4}{x^3}\\
\frac{dy}{dx}&=\frac{d}{dx}\left(\frac{4}{x^3}\right)\\
\frac{dy}{dx}&=\frac{d}{dx}4x^{-3}\\
\frac{dy}{dx}&=4\frac{d}{dx}x^{-3}\\\\
\because \frac{d}{dx}x^n&=nx^{n-1}\\\\
\therefore \frac{dy}{dx}&=4(-3x^{-3-1})\\
\frac{dy}{dx}&=-12x^{-4}\\
\frac{dy}{dx}&=\frac{-12}{x^4}\\
\frac{dy}{dx}&=-\frac{12}{x^4}\\
\end{align*}
Example 10:
$$ y=12x+\frac{3}{x},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
\begin{align*}
y&=12x+\frac{3}{x}\\
\frac{dy}{dx}&=\frac{d}{dx}\left(12x+\frac{3}{x}\right)\\
\frac{dy}{dx}&=\frac{d}{dx}12x+\frac{d}{dx}\frac{3}{x}\\
\frac{dy}{dx}&=\frac{d}{dx}12x+\frac{d}{dx}3x^{-1}\\
\frac{dy}{dx}&=12\frac{d}{dx}x+3\frac{d}{dx}x^{-1}\\\\
\because\;\; \frac{d}{dx}x&=1\;\; \textup{and} \;\:\frac{d}{dx}x^n=nx^{n-1}\\\\
\therefore\;\; \frac{dy}{dx}&=12(1)+3(-1x^{-1-1})\\
\frac{dy}{dx}&=12+3(-1x^{-2})\\
\frac{dy}{dx}&=12+3\left(\frac{-1}{x^2}\right)\\
\frac{dy}{dx}&=12-\frac{3}{x^2}\\
\end{align*}