derivative100 [41-50]

41. $\frac{d}{dx} (x)sqrt(4-x^2)$

$\begin{aligned} &\frac{d}{dx} x\sqrt{4-x^2}=\sqrt{4-x^2}+x\frac{-2x}{2\sqrt{4-x^2}}\\ &=\frac{4-2x^2}{\sqrt{4-x^2}} \end{aligned}$

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

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

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

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

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44. $\frac{d}{dx} cos(arcsinx)$

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

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45. $\frac{d}{dx} ln(x^2 + 3x + 5)$

$\begin{aligned} &\frac{d}{dx} ln(x^2 + 3x + 5)\\ &=\frac{2x+3}{x^2 + 3x + 5} \end{aligned}$

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46. $\frac{d}{dx}(arctan(4x))^2$

$\begin{aligned} &\frac{d}{dx}(arctan(4x))^2\\ &=2arctan(4x)\frac{4}{1+16x^2}\\ &=\frac{8\tan^{-1}{4x}}{1+16x^2} \end{aligned}$

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47. $\frac{d}{dx}cubert(x^2)$

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

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48. $\frac{d}{dx}sin(sqrt(x) lnx)$

$\begin{aligned} &\frac{d}{dx}sin(\sqrt{x} lnx)\\ &=cos(\sqrt{x} lnx)(\frac{1}{2\sqrt x}lnx+\sqrt x \frac{1}{x})\\ &=cos(\sqrt{x} lnx)\frac{ln(x)\sqrt{x}+2\sqrt x}{2x} \end{aligned}$

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49. $\frac{d}{dx}csc(x^2)$

$\begin{aligned} &\frac{d}{dx}csc(x^2)=-csc(x^2)cot(x^2)2x\\ &=-2xcsc(x^2)cot(x^2)\\ &cf) \frac{d}{dx}csc^2x=2cscx(-cscxcotx)\\ &=-2csc^2xcotx\\ \end{aligned}$

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

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

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