derivatie100 [51-60]

51. $\frac{d}{dx}10^x$

$\begin{aligned} &\frac{d}{dx}10^x\\ &=10^xln(10) \end{aligned}$

$\begin{aligned} &y=a^x, lny=xlna \\ &(1/y)dy=(lna)dx \\ &dy/dx=ylna=a^xlna \\ or\\ &10^x=e^{ln10^x} \end{aligned}$

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52. $\frac{d}{dx}cubert(x+(lnx)^2)$

$\begin{aligned} &\frac{d}{dx}\sqrt[3]{x+(lnx)^2}\\ &=\frac{d}{dx}(x+(lnx)^2)^{1/3}=\frac{1}{3}(x+(lnx)^2)^{-2/3}(1+2ln(x)\frac{1}{x})\\ &=\frac{1}{3(x+(lnx)^2)^{2/3}}(1+\frac{2lnx}{x}) \end{aligned}$

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53. $\frac{d}{dx}x^{3/4} – 2x^{1/4}$

$\begin{aligned} &\frac{d}{dx}x^{3/4} – 2x^{1/4}\\ &=\frac{3}{4}x^{-1/4}-\frac{1}{2}x^{-3/4}\\ &=\frac{3x^{3/4}}{4x}-\frac{2x^{1/4}}{4x}\\ &=\frac{3\sqrt[4]x^3-2\sqrt[4]x}{4x} \end{aligned}$

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54. $\frac{d}{dx}log_2 (x \sqrt{1+x^2})$

$\begin{aligned} &\frac{d}{dx}log_2 (x \sqrt{1+x^2})\\ &((d/dx) log_ax=\frac{1}{xlna})\\ &=\frac{\sqrt{1+x^2}+x\frac{2x}{2\sqrt{1+x^2}}}{x\sqrt{1+x^2}ln2}\\ &=\frac{1+2x^2}{x(1+x^2)ln2} \end{aligned}$

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55. $\frac{d}{dx}(x-1)/(x^2-x+1)$

$\begin{aligned} &\frac{d}{dx} \frac{x-1}{x^2-x+1}\\ &=\frac{x^2-x+1-(x-1)(2x-1)}{(x^2-x+1)^2}\\ &=\frac{x^2-x+1-(2x^2-3x+1)}{(x^2-x+1)^2}\\ &=\frac{-x^2+2x}{(x^2-x+1)^2} \end{aligned}$

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56. $\frac{d}{dx}\frac{1}{3} cos^3x – cosx$

$\begin{aligned} &\frac{d}{dx} \frac{1}{3}cos^3x – cosx\\ &=\frac{1}{3}3cos^2x(-sinx)+sinx\\ &=sinx(1-cos^2x)=sin^3x \end{aligned}$

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57. $\frac{d}{dx}e^{xcosx}$

$\begin{aligned} &\frac{d}{dx}e^{xcosx}=e^{xcosx}(cosx-xsinx)\\ \end{aligned}$

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58. $\frac{d}{dx}(x-\sqrt{x})(x+\sqrt{x})$

$\begin{aligned} &\frac{d}{dx}(x-\sqrt{x})(x+\sqrt{x})=\frac{d}{dx}x^2-x\\ &=2x-1 \end{aligned}$

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59. $\frac{d}{dx}arccot(\frac{1}{x})$

$\begin{aligned} &\frac{d}{dx}arccot(\frac 1 x)\\ & y=arccot \frac 1 x, \frac 1 x=cot y, -x^{-2}dx=-csc^2ydy\\ & R.T., angle=y, o=x, a=1, h=\sqrt{x^2+1} \\ & \frac {dy}{dx}=\frac {x^{-2}}{csc^2 y}=\frac{1}{x^2\frac{1+x^2}{x^2}}=\frac{1}{1+x^2} \\ &(=\frac{d}{dx}arctan(x)) \end{aligned}$
Alt.
$\frac{d}{dx} arccot(x)=-\frac{1}{1+x^2}\\ \frac{d}{dx} arccot(1/x)=-\frac{1}{1+1/x^2}(-1/x^2)\\ =\frac{1}{x^2+1}$

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60. $\frac{d}{dx}(x)(arctanx) – ln(\sqrt{x^2+1})$

$\begin{aligned} &\frac{d}{dx}(x)(arctanx) – ln(\sqrt{x^2+1})\\ &=arctanx+x\frac{1}{1+x^2}-\frac{1}{\sqrt{x^2+1}}\frac{1}{2\sqrt{x^2+1}}2x\\ &=arctanx+\frac{x}{1+x^2}-\frac{x}{1+x^2}\\ &=arctan x \end{aligned}$

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