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Quotient Rule

Calculate $\frac{d}{dx}\left[\frac{\log\log x}{x}\right]$ .

$8x\ln x+4x$

$\frac{x}{x^2}$

Calculate $\frac{d}{dx}\left[\frac{\left(x^2\right)}{\left(x+1\right)}\right]$ .

$\frac{x^2+x}{\left(x+1\right)^2}$

$\frac{x^2+2x}{\left(x+1\right)^2}$

$\frac{x^2+2}{\left(x+1\right)^2}$

Calculate $\frac{d}{dx}\left[\frac{2x-3}{2x+1}\right]$ .

$\frac{4}{\left(2x+1\right)^2}$

$\frac{8}{\left(x+1\right)^2}$

$\frac{8}{\left(2x+1\right)^2}$

Calculate $\frac{d}{dx}\left[\frac{\sqrt{1+x}}{\sqrt{1-x}}\right]$ .

$\frac{5}{\left(1-x\right)\sqrt{1-x^2}}$

$\frac{1}{\left(1-x\right)\sqrt{1-x^2}}$

$\frac{2}{\left(1-x\right)\sqrt{1-x^2}}$

Calculate $\frac{d}{dx}\left[\frac{\left(x+5\right)}{\left(5-x\right)}\right]$ .

$\frac{20}{\left(20-x\right)^2}$

$\frac{10}{\left(5-x\right)^2}$

$\frac{10}{\left(20-x\right)^2}$

Calculate $\frac{d}{dx}\left[\frac{\left(x+5\right)}{e^x}\right]$ .

$2e^x$

$\frac{e^{x\ }+\left(x+5\right)e^x}{e^{2x}}$

$\frac{e^{x\ }-\left(x+5\right)e^x}{e^{2x}}$

If $f\left(x\right)=x^2$ and $g\left(x\right)=4$ .Calculate $\frac{d}{dx}\left[\frac{\left(x^2\right)}{4}\right]$ .

$2x$

$2x+2$

$\frac{x}{2}$

Calculate $\frac{d}{dx}\left[\frac{\left(x+a\right)}{\left(a-x\right)}\right]$ .

$\frac{2a}{\left(a-x\right)^2}$

$\frac{2a}{\left(a+x\right)^2}$

$\frac{a}{\left(a-x\right)^2}$

Calculate $\frac{d}{dx}\left[\frac{\cos\left(x\right)}{x}\right]$ .

$\frac{-x\sin x+\cos x}{x^2}$

$\frac{x\sin x+\cos x}{x^2}$

$-\sin x+\frac{1}{x}$

Calculate $\frac{d}{dx}\left[\frac{\left(x+2\right)}{\left(2-x\right)}\right]$ .

$\frac{2}{\left(2-x\right)^2}$

$\frac{2}{\left(2+x\right)^2}$

$\frac{4}{\left(2-x\right)^2}$

Questions

Time

Score

Quotient Rule

Calculate ﻿ddx[log⁡log⁡xx]\frac{d}{dx}\left[\frac{\log\log x}{x}\right]dxd​[xloglogx​]﻿.

﻿8xln⁡x+4x8x\ln x+4x8xlnx+4x﻿

﻿xx2\frac{x}{x^2}x2x​﻿

﻿xx2\frac{x}{x^2}x2x​﻿

Calculate ﻿ddx[(x2)(x+1)]\frac{d}{dx}\left[\frac{\left(x^2\right)}{\left(x+1\right)}\right]dxd​[(x+1)(x2)​]﻿.

﻿x2+x(x+1)2\frac{x^2+x}{\left(x+1\right)^2}(x+1)2x2+x​﻿

﻿x2+2x(x+1)2\frac{x^2+2x}{\left(x+1\right)^2}(x+1)2x2+2x​﻿

﻿x2+2(x+1)2\frac{x^2+2}{\left(x+1\right)^2}(x+1)2x2+2​﻿

Calculate ﻿ddx[2x−32x+1]\frac{d}{dx}\left[\frac{2x-3}{2x+1}\right]dxd​[2x+12x−3​]﻿.

﻿4(2x+1)2\frac{4}{\left(2x+1\right)^2}(2x+1)24​﻿

﻿8(x+1)2\frac{8}{\left(x+1\right)^2}(x+1)28​﻿

﻿8(2x+1)2\frac{8}{\left(2x+1\right)^2}(2x+1)28​﻿

Calculate ﻿ddx[1+x1−x]\frac{d}{dx}\left[\frac{\sqrt{1+x}}{\sqrt{1-x}}\right]dxd​[1−x​1+x​​]﻿.

﻿5(1−x)1−x2\frac{5}{\left(1-x\right)\sqrt{1-x^2}}(1−x)1−x2​5​﻿

﻿1(1−x)1−x2\frac{1}{\left(1-x\right)\sqrt{1-x^2}}(1−x)1−x2​1​﻿

﻿2(1−x)1−x2\frac{2}{\left(1-x\right)\sqrt{1-x^2}}(1−x)1−x2​2​﻿

Calculate ﻿ddx[(x+5)(5−x)]\frac{d}{dx}\left[\frac{\left(x+5\right)}{\left(5-x\right)}\right]dxd​[(5−x)(x+5)​]﻿.

﻿20(20−x)2\frac{20}{\left(20-x\right)^2}(20−x)220​﻿

﻿10(5−x)2\frac{10}{\left(5-x\right)^2}(5−x)210​﻿

﻿10(20−x)2\frac{10}{\left(20-x\right)^2}(20−x)210​﻿

Calculate ﻿ddx[(x+5)ex]\frac{d}{dx}\left[\frac{\left(x+5\right)}{e^x}\right]dxd​[ex(x+5)​]﻿.

﻿2ex2e^x2ex﻿

﻿ex +(x+5)exe2x\frac{e^{x\ }+\left(x+5\right)e^x}{e^{2x}}e2xex +(x+5)ex​﻿

﻿ex −(x+5)exe2x\frac{e^{x\ }-\left(x+5\right)e^x}{e^{2x}}e2xex −(x+5)ex​﻿

If ﻿f(x)=x2f\left(x\right)=x^2f(x)=x2﻿ and ﻿g(x)=4g\left(x\right)=4g(x)=4﻿.Calculate ﻿ddx[(x2)4]\frac{d}{dx}\left[\frac{\left(x^2\right)}{4}\right]dxd​[4(x2)​]﻿.

﻿2x2x2x﻿

﻿2x+22x+22x+2﻿

﻿x2\frac{x}{2}2x​﻿

Calculate ﻿ddx[(x+a)(a−x)]\frac{d}{dx}\left[\frac{\left(x+a\right)}{\left(a-x\right)}\right]dxd​[(a−x)(x+a)​]﻿.

﻿2a(a−x)2\frac{2a}{\left(a-x\right)^2}(a−x)22a​﻿

﻿2a(a+x)2\frac{2a}{\left(a+x\right)^2}(a+x)22a​﻿

﻿a(a−x)2\frac{a}{\left(a-x\right)^2}(a−x)2a​﻿

Calculate ﻿ddx[cos⁡(x)x]\frac{d}{dx}\left[\frac{\cos\left(x\right)}{x}\right]dxd​[xcos(x)​]﻿.

﻿−xsin⁡x+cos⁡xx2\frac{-x\sin x+\cos x}{x^2}x2−xsinx+cosx​﻿

﻿xsin⁡x+cos⁡xx2\frac{x\sin x+\cos x}{x^2}x2xsinx+cosx​﻿

﻿−sin⁡x+1x-\sin x+\frac{1}{x}−sinx+x1​﻿

Calculate ﻿ddx[(x+2)(2−x)]\frac{d}{dx}\left[\frac{\left(x+2\right)}{\left(2-x\right)}\right]dxd​[(2−x)(x+2)​]﻿.

﻿2(2−x)2\frac{2}{\left(2-x\right)^2}(2−x)22​﻿

﻿2(2+x)2\frac{2}{\left(2+x\right)^2}(2+x)22​﻿

﻿4(2−x)2\frac{4}{\left(2-x\right)^2}(2−x)24​﻿

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Calculate $\frac{d}{dx}\left[\frac{\log\log x}{x}\right]$ .

$8x\ln x+4x$

$\frac{x}{x^2}$

$\frac{x}{x^2}$

Calculate $\frac{d}{dx}\left[\frac{\left(x^2\right)}{\left(x+1\right)}\right]$ .

$\frac{x^2+x}{\left(x+1\right)^2}$

$\frac{x^2+2x}{\left(x+1\right)^2}$

$\frac{x^2+2}{\left(x+1\right)^2}$

Calculate $\frac{d}{dx}\left[\frac{2x-3}{2x+1}\right]$ .

$\frac{4}{\left(2x+1\right)^2}$

$\frac{8}{\left(x+1\right)^2}$

$\frac{8}{\left(2x+1\right)^2}$

Calculate $\frac{d}{dx}\left[\frac{\sqrt{1+x}}{\sqrt{1-x}}\right]$ .

$\frac{5}{\left(1-x\right)\sqrt{1-x^2}}$

$\frac{1}{\left(1-x\right)\sqrt{1-x^2}}$

$\frac{2}{\left(1-x\right)\sqrt{1-x^2}}$

Calculate $\frac{d}{dx}\left[\frac{\left(x+5\right)}{\left(5-x\right)}\right]$ .

$\frac{20}{\left(20-x\right)^2}$

$\frac{10}{\left(5-x\right)^2}$

$\frac{10}{\left(20-x\right)^2}$

Calculate $\frac{d}{dx}\left[\frac{\left(x+5\right)}{e^x}\right]$ .

$2e^x$

$\frac{e^{x\ }+\left(x+5\right)e^x}{e^{2x}}$

$\frac{e^{x\ }-\left(x+5\right)e^x}{e^{2x}}$

If $f\left(x\right)=x^2$ and $g\left(x\right)=4$ .Calculate $\frac{d}{dx}\left[\frac{\left(x^2\right)}{4}\right]$ .

$2x$

$2x+2$

$\frac{x}{2}$

Calculate $\frac{d}{dx}\left[\frac{\left(x+a\right)}{\left(a-x\right)}\right]$ .

$\frac{2a}{\left(a-x\right)^2}$

$\frac{2a}{\left(a+x\right)^2}$

$\frac{a}{\left(a-x\right)^2}$

Calculate $\frac{d}{dx}\left[\frac{\cos\left(x\right)}{x}\right]$ .

$\frac{-x\sin x+\cos x}{x^2}$

$\frac{x\sin x+\cos x}{x^2}$

$-\sin x+\frac{1}{x}$

Calculate $\frac{d}{dx}\left[\frac{\left(x+2\right)}{\left(2-x\right)}\right]$ .

$\frac{2}{\left(2-x\right)^2}$

$\frac{2}{\left(2+x\right)^2}$

$\frac{4}{\left(2-x\right)^2}$