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Differentiation of Logarithmic Functions

Fill in the blank:
$\frac{d}{dx}\left[\sqrt{\ln x}\right]=$ __________.

$\frac{1+2\ln\ln x}{2\sqrt{\ln x}}$

$\ln\ln\frac{1+2\ln\ln x}{2\sqrt{\ln x}}$

$\frac{1}{2\sqrt{\ln x}}$

Fill in the blank:
$\frac{d}{dx}\left[x\ln x\right]=$ __________.

$-\frac{3x^x}{1-x^2}$

$\ln\ln x+1$

$x^x\left(\log\log x+1\right)$

Fill in the blank:
$\frac{d}{dx}\left[\left(x^2+5x\right)\right]=$ __________.

$\frac{2x+5}{x^2+5x}$

$\frac{2x+5}{\sqrt{1-x^2}}$

$\frac{1}{\sqrt{1+x^2}}$

Fill in the blank:
$\frac{d}{dx}\left[5x\sqrt{\ln x}\right]=$ __________.

$5\left(\sqrt{\ln x}+\frac{1}{2\sqrt{\ln x}}\right)$

$\frac{5+2\ln\ln x}{2\sqrt{\ln x}}$

$5x\ln\ln x+\frac{x}{2}$

Fill in the blank:
If $f\left(x\right)=\ln\ln\left(9x^2+1\right)$ . Then $\frac{d}{dx}\left[f\left(x\right)\right]=$ __________.

$\frac{3}{9x^2+1}$

$\frac{1}{9x^2+1}$

$\frac{18x}{9x^2+1}$

Fill in the blank:
$\frac{d}{dx}\left[\ln\ln\left(\frac{x^2+1}{5}\right)\right]=$ __________.

$\frac{2x}{-1+x^4}$

$\frac{2x}{x^2+1}$

$\frac{-1}{\sqrt{1-x^2}}$

Fill in the blank:
$\frac{d}{dx}\left[\left(\tan\tan x\right)\right]=$ __________.

$2\cos\cos2x$

$2\cot\cot2x$

$2\csc\csc2x$

Fill in the blank:
$\frac{d}{dx}\left[\ln\ln\left(x^2+3\right)\right]=$ __________.

$\frac{2x}{3-x^2}$

$\frac{1}{\sqrt{9-x^2}}$

$\frac{2x}{x^2+3}$

Fill in the blank:
$\frac{d}{dx}\left[\ln\left(\sin x\right)\right]=$ __________.

$\cot\cot x$

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

$2\csc x$

Fill in the blank:
$\frac{d}{dx}\left[\frac{1}{3}\ln\ln x^3\right]=$ __________.

$\frac{1}{\sqrt{2+x^2}}$

$\frac{1}{x}$

$\frac{3}{x}$

Questions

Time

Score

Differentiation of Logarithmic Functions

Fill in the blank:﻿ddx[ln⁡x]=\frac{d}{dx}\left[\sqrt{\ln x}\right]=dxd​[lnx​]=﻿ __________.

﻿1+2ln⁡ln⁡x2ln⁡x\frac{1+2\ln\ln x}{2\sqrt{\ln x}}2lnx​1+2lnlnx​﻿

﻿ln⁡ln⁡1+2ln⁡ln⁡x2ln⁡x\ln\ln\frac{1+2\ln\ln x}{2\sqrt{\ln x}}lnln2lnx​1+2lnlnx​﻿

﻿12ln⁡x\frac{1}{2\sqrt{\ln x}}2lnx​1​﻿

Fill in the blank:﻿ddx[xln⁡x]=\frac{d}{dx}\left[x\ln x\right]=dxd​[xlnx]=﻿ __________.

﻿−3xx1−x2-\frac{3x^x}{1-x^2}−1−x23xx​﻿

﻿ln⁡ln⁡x+1\ln\ln x+1lnlnx+1﻿

﻿xx(log⁡log⁡x+1)x^x\left(\log\log x+1\right)xx(loglogx+1)﻿

Fill in the blank:﻿ddx[(x2+5x)]=\frac{d}{dx}\left[\left(x^2+5x\right)\right]=dxd​[(x2+5x)]=﻿ __________.

﻿2x+5x2+5x\frac{2x+5}{x^2+5x}x2+5x2x+5​﻿

﻿2x+51−x2\frac{2x+5}{\sqrt{1-x^2}}1−x2​2x+5​﻿

﻿11+x2\frac{1}{\sqrt{1+x^2}}1+x2​1​﻿

Fill in the blank:﻿ddx[5xln⁡x]=\frac{d}{dx}\left[5x\sqrt{\ln x}\right]=dxd​[5xlnx​]=﻿ __________.

﻿5(ln⁡x+12ln⁡x)5\left(\sqrt{\ln x}+\frac{1}{2\sqrt{\ln x}}\right)5(lnx​+2lnx​1​)﻿

﻿5+2ln⁡ln⁡x2ln⁡x\frac{5+2\ln\ln x}{2\sqrt{\ln x}}2lnx​5+2lnlnx​﻿

﻿5xln⁡ln⁡x+x25x\ln\ln x+\frac{x}{2}5xlnlnx+2x​﻿

Fill in the blank:If ﻿f(x)=ln⁡ln⁡(9x2+1)f\left(x\right)=\ln\ln\left(9x^2+1\right)f(x)=lnln(9x2+1)﻿. Then ﻿ddx[f(x)]=\frac{d}{dx}\left[f\left(x\right)\right]=dxd​[f(x)]=﻿ __________.

﻿39x2+1\frac{3}{9x^2+1}9x2+13​﻿

﻿19x2+1\frac{1}{9x^2+1}9x2+11​﻿

﻿18x9x2+1\frac{18x}{9x^2+1}9x2+118x​﻿

Fill in the blank:﻿ddx[ln⁡ln⁡(x2+15)]=\frac{d}{dx}\left[\ln\ln\left(\frac{x^2+1}{5}\right)\right]=dxd​[lnln(5x2+1​)]=﻿ __________.

﻿2x−1+x4\frac{2x}{-1+x^4}−1+x42x​﻿

﻿2xx2+1\frac{2x}{x^2+1}x2+12x​﻿

﻿−11−x2\frac{-1}{\sqrt{1-x^2}}1−x2​−1​﻿

Fill in the blank:﻿ddx[(tan⁡tan⁡x)]=\frac{d}{dx}\left[\left(\tan\tan x\right)\right]=dxd​[(tantanx)]=﻿ __________.

﻿2cos⁡cos⁡2x2\cos\cos2x2coscos2x﻿

﻿2cot⁡cot⁡2x2\cot\cot2x2cotcot2x﻿

﻿2csc⁡csc⁡2x2\csc\csc2x2csccsc2x﻿

Fill in the blank:﻿ddx[ln⁡ln⁡(x2+3)]=\frac{d}{dx}\left[\ln\ln\left(x^2+3\right)\right]=dxd​[lnln(x2+3)]=﻿ __________.

﻿2x3−x2\frac{2x}{3-x^2}3−x22x​﻿

﻿19−x2\frac{1}{\sqrt{9-x^2}}9−x2​1​﻿

﻿2xx2+3\frac{2x}{x^2+3}x2+32x​﻿

Fill in the blank:﻿ddx[ln⁡(sin⁡x)]=\frac{d}{dx}\left[\ln\left(\sin x\right)\right]=dxd​[ln(sinx)]=﻿ __________.

﻿cot⁡cot⁡x\cot\cot xcotcotx﻿

﻿sin⁡x1+cos⁡x\frac{\sin x}{1+\cos x}1+cosxsinx​﻿

﻿2csc⁡x2\csc x2cscx﻿

Fill in the blank:﻿ddx[13ln⁡ln⁡x3]=\frac{d}{dx}\left[\frac{1}{3}\ln\ln x^3\right]=dxd​[31​lnlnx3]=﻿ __________.

﻿12+x2\frac{1}{\sqrt{2+x^2}}2+x2​1​﻿

﻿1x\frac{1}{x}x1​﻿

﻿3x\frac{3}{x}x3​﻿

Incorrect

Great!

Fill in the blank:
$\frac{d}{dx}\left[\sqrt{\ln x}\right]=$ __________.

$\frac{1+2\ln\ln x}{2\sqrt{\ln x}}$

$\ln\ln\frac{1+2\ln\ln x}{2\sqrt{\ln x}}$

$\frac{1}{2\sqrt{\ln x}}$

Fill in the blank:
$\frac{d}{dx}\left[x\ln x\right]=$ __________.

$-\frac{3x^x}{1-x^2}$

$\ln\ln x+1$

$x^x\left(\log\log x+1\right)$

Fill in the blank:
$\frac{d}{dx}\left[\left(x^2+5x\right)\right]=$ __________.

$\frac{2x+5}{x^2+5x}$

$\frac{2x+5}{\sqrt{1-x^2}}$

$\frac{1}{\sqrt{1+x^2}}$

Fill in the blank:
$\frac{d}{dx}\left[5x\sqrt{\ln x}\right]=$ __________.

$5\left(\sqrt{\ln x}+\frac{1}{2\sqrt{\ln x}}\right)$

$\frac{5+2\ln\ln x}{2\sqrt{\ln x}}$

$5x\ln\ln x+\frac{x}{2}$

Fill in the blank:
If $f\left(x\right)=\ln\ln\left(9x^2+1\right)$ . Then $\frac{d}{dx}\left[f\left(x\right)\right]=$ __________.

$\frac{3}{9x^2+1}$

$\frac{1}{9x^2+1}$

$\frac{18x}{9x^2+1}$

Fill in the blank:
$\frac{d}{dx}\left[\ln\ln\left(\frac{x^2+1}{5}\right)\right]=$ __________.

$\frac{2x}{-1+x^4}$

$\frac{2x}{x^2+1}$

$\frac{-1}{\sqrt{1-x^2}}$

Fill in the blank:
$\frac{d}{dx}\left[\left(\tan\tan x\right)\right]=$ __________.

$2\cos\cos2x$

$2\cot\cot2x$

$2\csc\csc2x$

Fill in the blank:
$\frac{d}{dx}\left[\ln\ln\left(x^2+3\right)\right]=$ __________.

$\frac{2x}{3-x^2}$

$\frac{1}{\sqrt{9-x^2}}$

$\frac{2x}{x^2+3}$

Fill in the blank:
$\frac{d}{dx}\left[\ln\left(\sin x\right)\right]=$ __________.

$\cot\cot x$

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

$2\csc x$

Fill in the blank:
$\frac{d}{dx}\left[\frac{1}{3}\ln\ln x^3\right]=$ __________.

$\frac{1}{\sqrt{2+x^2}}$

$\frac{1}{x}$

$\frac{3}{x}$