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The Derivative Function Of The Ratio Of Two Functions

Consider $q\left(x\right)=\frac{\tan x}{\cos x}$ , find $q'\left(x\right)$ .

$q'\left(x\right)=\frac{\sec^2x\cos x-\sin x\tan x}{\cos^2x}$

$q'\left(x\right)=\frac{\sec^2x\cos x+\sin x\tan x}{\cos^2x}$

$q'\left(x\right)=\frac{\sec^2x\cos x+\sin x\tan x}{\cos^{ }x}$

Given $m\left(x\right)=\frac{\cos x}{x}$ , determine $m'\left(x\right)$ .

$m'\left(x\right)=\frac{\sin x\cdot x-\cos x}{x^{ }}$

$m'\left(x\right)=\frac{\sin x\cdot x-\cos x}{x^2}$

$m'\left(x\right)=\frac{\sin x\cdot x+\cos x}{x^2}$

Given $r\left(x\right)=\frac{2x}{\sqrt{1-x^2}}$ , determine $r'\left(x\right)$ .

$r'\left(x\right)=\frac{2\sqrt{1-x^2}+2x\frac{2x}{2\sqrt{1-x^2}}}{1-x^{ }}$

$r'\left(x\right)=\frac{2\left(\sqrt{1-x^2}\right)-2x\cdot\frac{2x}{2\sqrt{1-x^2}}}{1-x^2}$

$r'\left(x\right)=\frac{2\left(\sqrt{1-x^2}\right)+2x\cdot\frac{2x}{2\sqrt{1-x^2}}}{1-x^2}$

If $p\left(x\right)=\frac{2x}{x^2+1}$ , find $p'\left(x\right)$ .

$p'\left(x\right)=\frac{-2x^2-1}{\left(x^2-1\right)^2}$

$p'\left(x\right)=\frac{-2x^2+1}{\left(x^2+1\right)^2}$

$p'\left(x\right)=\frac{-2x^2+1}{\left(x^2+1\right)^{ }}$

Consider $q\left(x\right)=\frac{\ln x}{x}$ , find $q'\left(x\right)$ .

$q'\left(x\right)=\frac{1-\ln x}{x^2}$

$q'\left(x\right)=\frac{1-\ln x}{2}$

$q'\left(x\right)=\frac{1+\ln x}{x^2}$

Determine the derivative of . $h\left(x\right)=\frac{x^2}{e^x}$ .

$h'\left(x\right)=\frac{2xe^x-x^2e^x}{\left(e^x\right)^2}$

$h'\left(x\right)=\frac{2xe^x+x^2e^x}{\left(e^x\right)^2}$

$h'\left(x\right)=\frac{2xe^x-x^2e^x}{\left(e^x\right)^{ }}$

Determine the derivative of $h\left(x\right)=\frac{\sqrt{x}}{e^x}$ .

$h'\left(x\right)=\frac{1-2x}{2\sqrt{x}e^x}$

$h'\left(x\right)=\frac{1}{2\sqrt{x}e^x}$

$h'\left(x\right)=\frac{1+2x}{2\sqrt{x}e^x}$

Consider $q\left(x\right)=\frac{\ln x}{x}$ , find $q'\left(x\right)$ .

$q'\left(x\right)=\frac{1-\ln x}{x^2}$

$q'\left(x\right)=\frac{1-\ln x}{2}$

$q'\left(x\right)=\frac{1+\ln x}{x^2}$

If $p\left(x\right)=\frac{e^x}{\ln x}$ , find $p'\left(x\right)$ .

$p'\left(x\right)=\frac{\frac{e^x}{x}-\ln x\cdot e^x}{\left(\ln x\right)^2}$

$p'\left(x\right)=\frac{\frac{e^x}{x}+\ln x\cdot e^x}{\left(\ln x\right)^2}$

$p'\left(x\right)=\frac{\frac{e^x}{x}-\ln x\cdot e^x}{\left(\ln x\right)^{ }}$

Given $m\left(x\right)=\frac{e^x}{\cos x}$ , determine $m'\left(x\right)$ .

$m'\left(x\right)=\frac{e^x\cdot\sin x+e^x\cos x}{\cos^2x}$

$m'\left(x\right)=\frac{e^x\cdot\sin x-e^x\cos x}{\cos^2x}$

$m'\left(x\right)=\frac{e^x\cdot\sin x-e^x\cos x}{\cos^{ }x}$

Questions

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The Derivative Function Of The Ratio Of Two Functions

Consider ﻿q(x)=tan⁡xcos⁡xq\left(x\right)=\frac{\tan x}{\cos x}q(x)=cosxtanx​﻿ , find ﻿q′(x)q'\left(x\right)q′(x)﻿ .

﻿q′(x)=sec⁡2xcos⁡x−sin⁡xtan⁡xcos⁡2xq'\left(x\right)=\frac{\sec^2x\cos x-\sin x\tan x}{\cos^2x}q′(x)=cos2xsec2xcosx−sinxtanx​﻿

﻿q′(x)=sec⁡2xcos⁡x+sin⁡xtan⁡xcos⁡2xq'\left(x\right)=\frac{\sec^2x\cos x+\sin x\tan x}{\cos^2x}q′(x)=cos2xsec2xcosx+sinxtanx​﻿

﻿q′(x)=sec⁡2xcos⁡x+sin⁡xtan⁡xcos⁡xq'\left(x\right)=\frac{\sec^2x\cos x+\sin x\tan x}{\cos^{ }x}q′(x)=cosxsec2xcosx+sinxtanx​﻿

Given ﻿m(x)=cos⁡xxm\left(x\right)=\frac{\cos x}{x}m(x)=xcosx​﻿ , determine ﻿m′(x)m'\left(x\right)m′(x)﻿ .

﻿m′(x)=sin⁡x⋅x−cos⁡xxm'\left(x\right)=\frac{\sin x\cdot x-\cos x}{x^{ }}m′(x)=xsinx⋅x−cosx​﻿

﻿m′(x)=sin⁡x⋅x−cos⁡xx2m'\left(x\right)=\frac{\sin x\cdot x-\cos x}{x^2}m′(x)=x2sinx⋅x−cosx​﻿

﻿m′(x)=sin⁡x⋅x+cos⁡xx2m'\left(x\right)=\frac{\sin x\cdot x+\cos x}{x^2}m′(x)=x2sinx⋅x+cosx​﻿

Given ﻿r(x)=2x1−x2r\left(x\right)=\frac{2x}{\sqrt{1-x^2}}r(x)=1−x2​2x​﻿ , determine ﻿r′(x)r'\left(x\right)r′(x)﻿ .

﻿r′(x)=21−x2+2x2x21−x21−xr'\left(x\right)=\frac{2\sqrt{1-x^2}+2x\frac{2x}{2\sqrt{1-x^2}}}{1-x^{ }}r′(x)=1−x21−x2​+2x21−x2​2x​​﻿

﻿r′(x)=2(1−x2)−2x⋅2x21−x21−x2r'\left(x\right)=\frac{2\left(\sqrt{1-x^2}\right)-2x\cdot\frac{2x}{2\sqrt{1-x^2}}}{1-x^2}r′(x)=1−x22(1−x2​)−2x⋅21−x2​2x​​﻿

﻿r′(x)=2(1−x2)+2x⋅2x21−x21−x2r'\left(x\right)=\frac{2\left(\sqrt{1-x^2}\right)+2x\cdot\frac{2x}{2\sqrt{1-x^2}}}{1-x^2}r′(x)=1−x22(1−x2​)+2x⋅21−x2​2x​​﻿

If ﻿p(x)=2xx2+1p\left(x\right)=\frac{2x}{x^2+1}p(x)=x2+12x​﻿ , find ﻿p′(x)p'\left(x\right)p′(x)﻿ .

﻿p′(x)=−2x2−1(x2−1)2p'\left(x\right)=\frac{-2x^2-1}{\left(x^2-1\right)^2}p′(x)=(x2−1)2−2x2−1​﻿

﻿p′(x)=−2x2+1(x2+1)2p'\left(x\right)=\frac{-2x^2+1}{\left(x^2+1\right)^2}p′(x)=(x2+1)2−2x2+1​﻿

﻿p′(x)=−2x2+1(x2+1)p'\left(x\right)=\frac{-2x^2+1}{\left(x^2+1\right)^{ }}p′(x)=(x2+1)−2x2+1​﻿

Consider ﻿q(x)=ln⁡xxq\left(x\right)=\frac{\ln x}{x}q(x)=xlnx​﻿ , find ﻿q′(x)q'\left(x\right)q′(x)﻿ .

﻿q′(x)=1−ln⁡xx2q'\left(x\right)=\frac{1-\ln x}{x^2}q′(x)=x21−lnx​﻿

﻿q′(x)=1−ln⁡x2q'\left(x\right)=\frac{1-\ln x}{2}q′(x)=21−lnx​﻿

﻿q′(x)=1+ln⁡xx2q'\left(x\right)=\frac{1+\ln x}{x^2}q′(x)=x21+lnx​﻿

Determine the derivative of .﻿h(x)=x2exh\left(x\right)=\frac{x^2}{e^x}h(x)=exx2​﻿ .

﻿h′(x)=2xex−x2ex(ex)2h'\left(x\right)=\frac{2xe^x-x^2e^x}{\left(e^x\right)^2}h′(x)=(ex)22xex−x2ex​﻿

﻿h′(x)=2xex+x2ex(ex)2h'\left(x\right)=\frac{2xe^x+x^2e^x}{\left(e^x\right)^2}h′(x)=(ex)22xex+x2ex​﻿

﻿h′(x)=2xex−x2ex(ex)h'\left(x\right)=\frac{2xe^x-x^2e^x}{\left(e^x\right)^{ }}h′(x)=(ex)2xex−x2ex​﻿

Determine the derivative of ﻿h(x)=xexh\left(x\right)=\frac{\sqrt{x}}{e^x}h(x)=exx​​﻿ .

﻿h′(x)=1−2x2xexh'\left(x\right)=\frac{1-2x}{2\sqrt{x}e^x}h′(x)=2x​ex1−2x​﻿

﻿h′(x)=12xexh'\left(x\right)=\frac{1}{2\sqrt{x}e^x}h′(x)=2x​ex1​﻿

﻿h′(x)=1+2x2xexh'\left(x\right)=\frac{1+2x}{2\sqrt{x}e^x}h′(x)=2x​ex1+2x​﻿

Consider ﻿q(x)=ln⁡xxq\left(x\right)=\frac{\ln x}{x}q(x)=xlnx​﻿ , find ﻿q′(x)q'\left(x\right)q′(x)﻿ .

﻿q′(x)=1−ln⁡xx2q'\left(x\right)=\frac{1-\ln x}{x^2}q′(x)=x21−lnx​﻿

﻿q′(x)=1−ln⁡x2q'\left(x\right)=\frac{1-\ln x}{2}q′(x)=21−lnx​﻿

﻿q′(x)=1+ln⁡xx2q'\left(x\right)=\frac{1+\ln x}{x^2}q′(x)=x21+lnx​﻿

If ﻿p(x)=exln⁡xp\left(x\right)=\frac{e^x}{\ln x}p(x)=lnxex​﻿ , find ﻿p′(x)p'\left(x\right)p′(x)﻿ .

﻿p′(x)=exx−ln⁡x⋅ex(ln⁡x)2p'\left(x\right)=\frac{\frac{e^x}{x}-\ln x\cdot e^x}{\left(\ln x\right)^2}p′(x)=(lnx)2xex​−lnx⋅ex​﻿

﻿p′(x)=exx+ln⁡x⋅ex(ln⁡x)2p'\left(x\right)=\frac{\frac{e^x}{x}+\ln x\cdot e^x}{\left(\ln x\right)^2}p′(x)=(lnx)2xex​+lnx⋅ex​﻿

﻿p′(x)=exx−ln⁡x⋅ex(ln⁡x)p'\left(x\right)=\frac{\frac{e^x}{x}-\ln x\cdot e^x}{\left(\ln x\right)^{ }}p′(x)=(lnx)xex​−lnx⋅ex​﻿

Given ﻿m(x)=excos⁡xm\left(x\right)=\frac{e^x}{\cos x}m(x)=cosxex​﻿ , determine ﻿m′(x)m'\left(x\right)m′(x)﻿ .

﻿m′(x)=ex⋅sin⁡x+excos⁡xcos⁡2xm'\left(x\right)=\frac{e^x\cdot\sin x+e^x\cos x}{\cos^2x}m′(x)=cos2xex⋅sinx+excosx​﻿

﻿m′(x)=ex⋅sin⁡x−excos⁡xcos⁡2xm'\left(x\right)=\frac{e^x\cdot\sin x-e^x\cos x}{\cos^2x}m′(x)=cos2xex⋅sinx−excosx​﻿

﻿m′(x)=ex⋅sin⁡x−excos⁡xcos⁡xm'\left(x\right)=\frac{e^x\cdot\sin x-e^x\cos x}{\cos^{ }x}m′(x)=cosxex⋅sinx−excosx​﻿

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Consider $q\left(x\right)=\frac{\tan x}{\cos x}$ , find $q'\left(x\right)$ .

$q'\left(x\right)=\frac{\sec^2x\cos x-\sin x\tan x}{\cos^2x}$

$q'\left(x\right)=\frac{\sec^2x\cos x+\sin x\tan x}{\cos^2x}$

$q'\left(x\right)=\frac{\sec^2x\cos x+\sin x\tan x}{\cos^{ }x}$

Given $m\left(x\right)=\frac{\cos x}{x}$ , determine $m'\left(x\right)$ .

$m'\left(x\right)=\frac{\sin x\cdot x-\cos x}{x^{ }}$

$m'\left(x\right)=\frac{\sin x\cdot x-\cos x}{x^2}$

$m'\left(x\right)=\frac{\sin x\cdot x+\cos x}{x^2}$

Given $r\left(x\right)=\frac{2x}{\sqrt{1-x^2}}$ , determine $r'\left(x\right)$ .

$r'\left(x\right)=\frac{2\sqrt{1-x^2}+2x\frac{2x}{2\sqrt{1-x^2}}}{1-x^{ }}$

$r'\left(x\right)=\frac{2\left(\sqrt{1-x^2}\right)-2x\cdot\frac{2x}{2\sqrt{1-x^2}}}{1-x^2}$

$r'\left(x\right)=\frac{2\left(\sqrt{1-x^2}\right)+2x\cdot\frac{2x}{2\sqrt{1-x^2}}}{1-x^2}$

If $p\left(x\right)=\frac{2x}{x^2+1}$ , find $p'\left(x\right)$ .

$p'\left(x\right)=\frac{-2x^2-1}{\left(x^2-1\right)^2}$

$p'\left(x\right)=\frac{-2x^2+1}{\left(x^2+1\right)^2}$

$p'\left(x\right)=\frac{-2x^2+1}{\left(x^2+1\right)^{ }}$

Consider $q\left(x\right)=\frac{\ln x}{x}$ , find $q'\left(x\right)$ .

$q'\left(x\right)=\frac{1-\ln x}{x^2}$

$q'\left(x\right)=\frac{1-\ln x}{2}$

$q'\left(x\right)=\frac{1+\ln x}{x^2}$

Determine the derivative of . $h\left(x\right)=\frac{x^2}{e^x}$ .

$h'\left(x\right)=\frac{2xe^x-x^2e^x}{\left(e^x\right)^2}$

$h'\left(x\right)=\frac{2xe^x+x^2e^x}{\left(e^x\right)^2}$

$h'\left(x\right)=\frac{2xe^x-x^2e^x}{\left(e^x\right)^{ }}$

Determine the derivative of $h\left(x\right)=\frac{\sqrt{x}}{e^x}$ .

$h'\left(x\right)=\frac{1-2x}{2\sqrt{x}e^x}$

$h'\left(x\right)=\frac{1}{2\sqrt{x}e^x}$

$h'\left(x\right)=\frac{1+2x}{2\sqrt{x}e^x}$

Consider $q\left(x\right)=\frac{\ln x}{x}$ , find $q'\left(x\right)$ .

$q'\left(x\right)=\frac{1-\ln x}{x^2}$

$q'\left(x\right)=\frac{1-\ln x}{2}$

$q'\left(x\right)=\frac{1+\ln x}{x^2}$

If $p\left(x\right)=\frac{e^x}{\ln x}$ , find $p'\left(x\right)$ .

$p'\left(x\right)=\frac{\frac{e^x}{x}-\ln x\cdot e^x}{\left(\ln x\right)^2}$

$p'\left(x\right)=\frac{\frac{e^x}{x}+\ln x\cdot e^x}{\left(\ln x\right)^2}$

$p'\left(x\right)=\frac{\frac{e^x}{x}-\ln x\cdot e^x}{\left(\ln x\right)^{ }}$

Given $m\left(x\right)=\frac{e^x}{\cos x}$ , determine $m'\left(x\right)$ .

$m'\left(x\right)=\frac{e^x\cdot\sin x+e^x\cos x}{\cos^2x}$

$m'\left(x\right)=\frac{e^x\cdot\sin x-e^x\cos x}{\cos^2x}$

$m'\left(x\right)=\frac{e^x\cdot\sin x-e^x\cos x}{\cos^{ }x}$