Basic Differentiation
Example 1:
$$ y=x^4,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=x^4$$
$$ \frac{dy}{dx}=\frac{d}{dx} x^4$$
$$ \because\;\; \frac{d}{dx} x^n=nx^{n-1}$$
$$ \therefore\;\; \frac{dy}{dx}=4x^{4-1}$$
$$ \frac{dy}{dx}=4x^3$$
Example 2:
$$ y=3x^5,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=3x^5$$
$$ \frac{dy}{dx}=\frac{d}{dx} 3x^5$$
$$ \frac{dy}{dx}=3\frac{d}{dx}x^5$$
$$ \because\;\; \frac{d}{dx} x^n=nx^{n-1}$$
$$ \therefore\;\; \frac{dy}{dx}=3(5x^{5-1})$$
$$ \frac{dy}{dx}=15x^4$$
Example 3:
$$ y=x^3+8,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=x^3+8$$
$$ \frac{dy}{dx}=\frac{d}{dx}(x^3+8)$$
$$ \frac{dy}{dx}=\frac{d}{dx}x^3+\frac{d}{dx}8$$
$$ \because\;\; \frac{d}{dx} x^n=nx^{n-1}\;\; \textup{and} \;\:\frac{d}{dx} c=0$$
$$ \therefore\;\; \frac{dy}{dx}=3x^{3-1}+0$$
$$ \frac{dy}{dx}=3x^2$$
Example 4:
$$ y=2x^2+5x,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=2x^2+5x$$
$$ \frac{dy}{dx}=\frac{d}{dx}(2x^2+5x)$$
$$ \frac{dy}{dx}=\frac{d}{dx}2x^2+\frac{d}{dx}5x$$
$$ \frac{dy}{dx}=2\frac{d}{dx}x^2+5\frac{d}{dx}x$$
$$ \because\;\; \frac{d}{dx}x^n=nx^{n-1}\;\; \textup{and} \;\:\frac{d}{dx}x=1$$
$$ \therefore\;\; \frac{dy}{dx}=2(2x^{2-1})+5(1)$$
$$ \frac{dy}{dx}=4x+5$$
Example 5:
$$ y=7x^6-9x^4+12x^2,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=7x^6-9x^4+12x^2$$
$$ \frac{dy}{dx}=\frac{d}{dx}(7x^6-9x^4+12x^2)$$
$$ \frac{dy}{dx}=\frac{d}{dx}7x^6-\frac{d}{dx}9x^4+\frac{d}{dx}12x^2$$
$$ \frac{dy}{dx}=7\frac{d}{dx}x^6-9\frac{d}{dx}x^4+12\frac{d}{dx}x^2$$
$$ \because\;\; \frac{d}{dx}x^n=nx^{n-1}$$
$$ \therefore\;\; \frac{dy}{dx}=7(6x^{6-1})-9(4x^{4-1})+12(2x^{2-1})$$
$$ \frac{dy}{dx}=42x^5-36x^3+24x$$
Example 6:
$$ y=3x^3-4x^2+7x-10,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=3x^3-4x^2+7x-10$$
$$ \frac{dy}{dx}=\frac{d}{dx}(3x^3-4x^2+7x-10)$$
$$ \frac{dy}{dx}=\frac{d}{dx}3x^3-\frac{d}{dx}4x^2+\frac{d}{dx}7x-\frac{d}{dx}10$$
$$ \frac{dy}{dx}=3\frac{d}{dx}x^3-4\frac{d}{dx}x^2+7\frac{d}{dx}x-\frac{d}{dx}10$$
$$ \because\;\; \frac{d}{dx}x^n=nx^{n-1}\,,\,\frac{d}{dx}x=1\,\textup{and}\,\frac{d}{dx}c=0$$
$$ \therefore\;\; \frac{dy}{dx}=3(3x^{3-1})-4(2x^{2-1})+7(1)-0$$
$$ \frac{dy}{dx}=9x^2-8x+7$$
Example 7:
$$ y=\frac{1}{x},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=\frac{1}{x}$$
$$ \frac{dy}{dx}=\frac{d}{dx}\left(\frac{1}{x}\right)$$
$$ \frac{dy}{dx}=\frac{d}{dx}x^{-1}$$
$$ \because \frac{d}{dx}x^n=nx^{n-1}$$
$$ \therefore \frac{dy}{dx}=-1x^{-1-1}$$
$$ \frac{dy}{dx}=-1x^{-2}$$
$$ \frac{dy}{dx}=\frac{-1}{x^2}$$
$$ \frac{dy}{dx}=-\frac{1}{x^2}$$
Example 8:
$$ y=\frac{1}{x^2},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=\frac{1}{x^2}$$
$$ \frac{dy}{dx}=\frac{d}{dx}\left(\frac{1}{x^2}\right)$$
$$ \frac{dy}{dx}=\frac{d}{dx}x^{-2}$$
$$ \because \frac{d}{dx}x^n=nx^{n-1}$$
$$ \therefore \frac{dy}{dx}=-2x^{-2-1}$$
$$ \frac{dy}{dx}=-2x^{-3}$$
$$ \frac{dy}{dx}=\frac{-2}{x^3}$$
$$ \frac{dy}{dx}=-\frac{2}{x^3}$$
Example 9:
$$ y=\frac{4}{x^3},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=\frac{4}{x^3}$$
$$ \frac{dy}{dx}=\frac{d}{dx}\left(\frac{4}{x^3}\right)$$
$$ \frac{dy}{dx}=\frac{d}{dx}4x^{-3}$$
$$ \frac{dy}{dx}=4\frac{d}{dx}x^{-3}$$
$$ \because \frac{d}{dx}x^n=nx^{n-1}$$
$$ \therefore \frac{dy}{dx}=4(-3x^{-3-1})$$
$$ \frac{dy}{dx}=-12x^{-4}$$
$$ \frac{dy}{dx}=\frac{-12}{x^4}$$
$$ \frac{dy}{dx}=-\frac{12}{x^4}$$
Example 10:
$$ y=12x+\frac{3}{x},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=12x+\frac{3}{x}$$
$$ \frac{dy}{dx}=\frac{d}{dx}\left(12x+\frac{3}{x}\right)$$
$$ \frac{dy}{dx}=\frac{d}{dx}12x+\frac{d}{dx}\frac{3}{x}$$
$$ \frac{dy}{dx}=\frac{d}{dx}12x+\frac{d}{dx}3x^{-1}$$
$$ \frac{dy}{dx}=12\frac{d}{dx}x+3\frac{d}{dx}x^{-1}$$
$$ \because\;\; \frac{d}{dx}x=1\;\; \textup{and} \;\:\frac{d}{dx}x^n=nx^{n-1}$$
$$ \therefore\;\; \frac{dy}{dx}=12(1)+3(-1x^{-1-1})$$
$$ \frac{dy}{dx}=12+3(-1x^{-2})$$
$$ \frac{dy}{dx}=12+3\left(\frac{-1}{x^2}\right)$$
$$ \frac{dy}{dx}=12-\frac{3}{x^2}$$
Example 11:
$$ y=4x^3-\frac{5}{x^2},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=4x^3-\frac{5}{x^2}$$
$$ \frac{dy}{dx}=\frac{d}{dx}\left(4x^3-\frac{5}{x^2}\right)$$
$$ \frac{dy}{dx}=\frac{d}{dx}4x^3-\frac{d}{dx}\frac{5}{x^2}$$
$$ \frac{dy}{dx}=\frac{d}{dx}4x^3-\frac{d}{dx}5x^{-2}$$
$$ \frac{dy}{dx}=4\frac{d}{dx}x^3-5\frac{d}{dx}x^{-2}$$
$$ \because\frac{d}{dx}x^n=nx^{n-1}$$
$$ \therefore\;\; \frac{dy}{dx}=4(3x^{3-1})-5(-2x^{-2-1})$$
$$ \frac{dy}{dx}=12x^2+10x^{-3}$$
$$ \frac{dy}{dx}=12x^2+\frac{10}{x^3}$$
Example 12:
$$ y=\frac{8}{x}+\frac{6}{x^2}+\frac{7}{x^3},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=\frac{8}{x}+\frac{6}{x^2}+\frac{7}{x^3}$$
$$ \frac{dy}{dx}=\frac{d}{dx}\left(\frac{8}{x}+\frac{6}{x^2}+\frac{7}{x^3},\right)$$
$$ \frac{dy}{dx}=\frac{d}{dx}\frac{8}{x}+\frac{d}{dx}\frac{6}{x^2}+\frac{d}{dx}\frac{7}{x^3},$$
$$ \frac{dy}{dx}=\frac{d}{dx}8x^{-1}+\frac{d}{dx}6x^{-2}+\frac{d}{dx}7x^{-3},$$
$$ \frac{dy}{dx}=8\frac{d}{dx}x^{-1}+6\frac{d}{dx}x^{-2}+7\frac{d}{dx}x^{-3},$$
$$ \because\frac{d}{dx}x^n=nx^{n-1}$$
$$ \therefore\;\; \frac{dy}{dx}=8(-1x^{-1-1})+6(-2x^{-2-1})+7(-3x^{-3-1})$$
$$ \frac{dy}{dx}=-8x^{-2}-12x^{-3}-21x^{-4}$$
$$ \frac{dy}{dx}=-\frac{8}{x^2}-\frac{12}{x^3}-\frac{21}{x^3}$$
Example 13:
$$ y=6x^4+\frac{3}{x^5},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=6x^4+\frac{3}{x^5}$$
$$ \frac{dy}{dx}=\frac{d}{dx}\left(6x^4+\frac{3}{x^5}\right)$$
$$ \frac{dy}{dx}=\frac{d}{dx}6x^4+\frac{d}{dx}\frac{3}{x^5}$$
$$ \frac{dy}{dx}=\frac{d}{dx}6x^4+\frac{d}{dx}3x^{-5}$$
$$ \frac{dy}{dx}=6\frac{d}{dx}x^4+3\frac{d}{dx}x^{-5}$$
$$ \because\frac{d}{dx}x^n=nx^{n-1}$$
$$ \therefore\;\; \frac{dy}{dx}=6(4x^{4-1})+3(-5x^{-5-1})$$
$$ \frac{dy}{dx}=24x^3-15x^{-6}$$
$$ \frac{dy}{dx}=24x^3-\frac{15}{x^6}$$
Example 14:
$$ y=\sqrt{x},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=\sqrt{x}$$
$$ \frac{dy}{dx}=\frac{d}{dx} \sqrt{x}$$
$$ \frac{dy}{dx}=\frac{d}{dx} x^{\frac{1}{2}}$$
$$ \because\;\; \frac{d}{dx} x^n=nx^{n-1}$$
$$ \therefore\;\; \frac{dy}{dx}=\frac{1}{2}x^{\frac{1}{2}-1}$$
$$ \frac{dy}{dx}=\frac{1}{2}x^{\frac{1-2}{2}}$$
$$ \frac{dy}{dx}=\frac{1}{2}x^{-\frac{1}{2}}$$
$$ \frac{dy}{dx}=\frac{1}{2x^{\frac{1}{2}}}$$
$$ \frac{dy}{dx}=\frac{1}{2\sqrt{x}}$$
Example 15:
$$ y=\frac{1}{\sqrt{x}},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$
Solution:
$$ y=\frac{1}{\sqrt{x}}$$
$$ \frac{dy}{dx}=\frac{d}{dx} \frac{1}{\sqrt{x}}$$
$$ \frac{dy}{dx}=\frac{d}{dx} \frac{1}{x^{\frac{1}{2}}}$$
$$ \frac{dy}{dx}=\frac{d}{dx} x^{-\frac{1}{2}}$$
$$ \because\;\; \frac{d}{dx} x^n=nx^{n-1}$$
$$ \therefore\;\; \frac{dy}{dx}=-\frac{1}{2}x^{-\frac{1}{2}-1}$$
$$ \frac{dy}{dx}=-\frac{1}{2}x^{\frac{-1-2}{2}}$$
$$ \frac{dy}{dx}=-\frac{1}{2}x^{-\frac{3}{2}}$$
$$ \frac{dy}{dx}=-\frac{1}{2x^{\frac{3}{2}}}$$
$$ \frac{dy}{dx}=-\frac{1}{2x^{1+\frac{1}{2}}}$$
$$ \frac{dy}{dx}=-\frac{1}{2x^1x^{\frac{1}{2}}}$$
$$ \frac{dy}{dx}=-\frac{1}{2x\sqrt{x}}$$