exercise - basic derivative rules

worked example: Find the derivative of $f(x)=4x^2$
$f^{\prime }(x)=4\cdot \frac{d}{dx}[x^2]$
$f^{\prime }(x)=4\cdot 2x$

worked example: Find the derivative of $f(x)=2x^7$
$f^{\prime }(x)=(7)(2)x^{7-1}=14x^6$

worked example: Find the derivative of $f(x)=4x^2+5$
$f^{\prime }(x)=\frac{d}{dx}[4x^2]+\frac{d}{dx}[5]$
$f^{\prime }(x)=8x+0$

worked example: Find the derivative of $f(x)=4x^2+5x$
$f^{\prime }(x)=\frac{d}{dx}[4x^2]+(5)\frac{d}{dx}[x]$
$f^{\prime }(x)=8x+(5)(1)$

worked example: Find the derivative of $f(x)=4x^2+5x+6$
$f^{\prime }(x)=\frac{d}{dx}[4x^2]+(5)\frac{d}{dx}[x]+\frac{d}{dx}[6]$
$f^{\prime }(x)=8x+5+0$

worked example: Find $f^{\prime }(1)$ for $f(x)=5x^3+2x+5$
$f^{\prime }(x)=15x^2+2$
$f^{\prime }(1)=17$

worked example: Differentiate $f(x)=x^{-\frac{1}{2}}$

$f^{\prime }(x)=-\frac{1}{2}\cdot x^{-\frac{3}{2}}$
$f^{\prime }(x)=-\frac{1}{2}\cdot \frac{1}{(\sqrt{x}{)}^3}$

worked example: Differentiate $f(x)=\frac{1}{2x}+x^3$

$f^{\prime }(x)=(\frac{1}{2})\frac{d}{dx}[x]+\frac{d}{dx}[x^2]$
$f^{\prime }(x)=\frac{1}{2}+2x$

worked example: Differentiate $f(x)=\frac{1}{x}+\frac{1}{\sqrt{x}}$

$f(x)=x^{-1}+x^{-\frac{1}{2}}$

$f^{\prime }(x)=(-1)x^{-2}+(-\frac{1}{2})x^{-\frac{3}{2}}$

worked example: Differentiate $f(x)=2x^2-\frac{1}{x}$
$f(x)=2x^2-x^{-1}$
$f^{\prime }(x)=4x+\frac{1}{x^2}$

worked example: Differentiate $f(x)=\frac{x-3}{2x^2-7x+3}$ by factorizing first

$f(x)=\frac{x-3}{2x^2-6x-x+3}$

$f(x)=\frac{x-3}{2x(x-3)+(-1)(x-3)}$

$f(x)=\frac{x-3}{(2x-1)(x-3)}$

$f(x)=\frac{1}{2x-1}$

$f^{\prime }(x)=(-1)(2x-1{)}^{-2}$

$f^{\prime }(x)=\frac{-1}{(2x-1{)}^2}$

worked example: Differentiate $f(x)=\frac{x-2}{3x^2-5x-2}$ by factorizing first

$f(x)=\frac{x-2}{3x^2-6x+x-2}$

$f(x)=\frac{x-2}{3x(x-2)+1(x-2)}$

$f(x)=\frac{x-2}{(3x+1)(x-2)}$

$f(x)=\frac{1}{3x+1}$

$f^{\prime }(x)=(-1)(3x+1{)}^{-2}$

$f^{\prime }(x)=\frac{-1}{(3x+1{)}^2}$

worked example: Differentiate $f(x)=e^{5x}+e^7-e^{-x}$

$f^{\prime }(x)=5e^{5x}+0+e^{-x}$

worked example: Differentiate $f(x)=-\frac{1}{2\sqrt{x}}$

$f(x)=-\frac{1}{2}{x}^{-\frac{1}{2}}$

$f^{\prime }(x)=\frac{1}{4}{x}^{-\frac{3}{2}}$

worked example: Differentiate $f(x)=\frac{e^{2x}-e^x+1}{e^x}$

$f(x)=\frac{e^{2x}}{e^x}-1+\frac{1}{e^x}$

$f(x)=e^x-1+\frac{1}{e^x}$

$f(x)=e^x-1+e^{-x}$

$f^{\prime }(x)=e^x-e^{-x}$

worked example: Differentiate $f(x)=\frac{1}{x^3}-\frac{2}{x^2}$

$f(x)=x^{-3}-2\cdot x^{-2}$
$f^{\prime }(x)=-3x^{-4}+4x^{-3}$