Calculus/Differentiation:Solutions

Set One: Find The Derivative By Definiton

1. 2x

$f(x)$	= $x^2$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{(x+\mathrm{\Delta }x{)}^2-x^2}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{x^2+2x\mathrm{\Delta }x+\mathrm{\Delta }{x}^2-x^2}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{2x\mathrm{\Delta }x+\mathrm{\Delta }{x}^2}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }2x+\mathrm{\Delta }x$
	= $2x$

2. 2

$f(x)$	= $2x+2$
$f^{\prime }(x)$	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{[2(x+\mathrm{\Delta }x)+2]-(2x+2)}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{2x+2\mathrm{\Delta }x+2-2x-2}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{2\mathrm{\Delta }x}{\mathrm{\Delta }x}$
	= $2$

3. x

$f(x)$	= $\frac{1}{2}{x}^2$
$f^{\prime }(x)$	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\frac{1}{2}(x+\mathrm{\Delta }x{)}^2-\frac{1}{2}{x}^2}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\frac{1}{2}(x^2+2x\mathrm{\Delta }x+\mathrm{\Delta }{x}^2)-\frac{1}{2}{x}^2}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\frac{x^2}{2}+\frac{2x\mathrm{\Delta }x}{2}+\frac{\mathrm{\Delta }{x}^2}{2}-\frac{x^2}{2}}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{2x\mathrm{\Delta }x+\mathrm{\Delta }{x}^2}{2\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }x+\mathrm{\Delta }x$
	= $x$

4. 4x + 4

$f(x)$	= $2x^2+4x+4$
$f^{\prime }(x)$	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{[2(x+\mathrm{\Delta }x{)}^2+4(x+\mathrm{\Delta }x)+4]-(2x^2+4x+4)}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{2(x^2+2x\mathrm{\Delta }x+\mathrm{\Delta }{x}^2)+4x+4\mathrm{\Delta }x+4-2x^2-4x-4}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{2x^2+4x\mathrm{\Delta }x+2\mathrm{\Delta }{x}^2+4\mathrm{\Delta }x-2x^2}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{4x\mathrm{\Delta }x+2\mathrm{\Delta }{x}^2+4\mathrm{\Delta }x}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }4x+2\mathrm{\Delta }x+4$
	= $4x+4$

Set Two: Find The Derivative By Rules

1. $4x$
2. $\frac{1}{\sqrt[3]{x^2}}$
3. $\frac{1}{\sqrt[3]{x^2}}-\frac{2}{x^3}$
4. $\frac{1}{x}-2e^x+\frac{1}{2\sqrt{x}}$
5. $\cos (x)-\sin (x)$
6. $2(x+5)$
7. $\frac{2x}{2\sqrt{1+x^2}}$
8. $\frac{x^3}{\sqrt{y}}+2\sqrt{y}3x^2$
9. $\frac{-8\sqrt{x}}{y^5}+\frac{1}{y^4\sqrt{x}}$
10. $2\sqrt{2x^2+1}(3y^4+2y)(12y^3+2)+\frac{2x(3y^4+2y)}{\sqrt{2x^2+1}}$
11. $\ln (4)4^x$
12. $\frac{2^{x-3}9x^2}{2\sqrt{x^3-2}}+3\ln (2)\sqrt{x^3-2}(2^{x-3})+\frac{1}{x}$
13. $4x{e}^x{y}^2{z}^3+2x{e}^{x}y{z}^4+(x{e}^x{y}^2{z}^4+e^x{y}^2{z}^4)$
14. $\frac{1}{x\ln 4}+\frac{2}{x}$
15. $3e^x+4\sin (x)-\frac{1}{4x}$

Set Three: Higher Order Derivatives

[Solutions Here]