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The Derivative Function Of The Product Of Two Functions

Calculate the derivative of $s\left(x\right)=x\cdot e^x\cdot\sin x$ .

$s'\left(x\right)=e^x\cdot\sin x+x\cdot e^x\cdot\cos x-x\cdot e^x\sin x$

$s'\left(x\right)=e^x\cdot\sin x-x\cdot e^x\cdot\cos x-x\cdot e^x\sin x$

$s'\left(x\right)=e^x\cdot\sin x+x\cdot e^x\cdot\cos x+x\cdot e^x\sin x$

Consider $w\left(x\right)=x\cdot\cos x\cdot\ln x$ . Find $w'\left(x\right)$ .

$w'\left(x\right)=\cos x\cdot\ln x-x\cdot\sin x\cdot\ln x+x\cdot\cos x\cdot\frac{1}{x}$

$w'\left(x\right)=\cos x\cdot\ln x-x\cdot\sin x\cdot\ln x-x\cdot\cos x\cdot\frac{1}{x}$

$w'\left(x\right)=\cos x\cdot\ln x+x\cdot\sin x\cdot\ln x+x\cdot\cos x\cdot\frac{1}{x}$

If $v\left(x\right)=\sin x\cdot e^x$ , calculate $v'\left(x\right)$ .

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

$v'\left(x\right)=\cos x\cdot e^x-\sin x\cdot e^x$

$v'\left(x\right)=\cos x\cdot e^x+\sin x\cdot e^x$

Consider $q\left(x\right)=\frac{1}{x}\cdot e^x$ , find $q'\left(x\right)$ .

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

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

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

Find the derivative of $t\left(x\right)=\ln x\cdot e^{-x}$ .

$t'\left(x\right)=\frac{1}{x}\cdot e^{-x}-\ln x\cdot e^{-x}$

$t'\left(x\right)=\frac{1}{x}\cdot e^{-x}+\ln x\cdot e^{-x}$

$t'\left(x\right)=\frac{1}{x}\cdot e^{-x}-\ln x\cdot e^x$

Given $u\left(x\right)=\sqrt{x}\cdot\tan x$ , determine $u'\left(x\right)$ .

$u'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\tan x+\sqrt{x}\cdot\sec^2x$

$u'\left(x\right)=12\sqrt{x}\cdot\tan x+\sqrt{x}\cdot\sec^2x$

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

Let $m\left(x\right)=x^3\cdot\ln x$ , calculate $m'\left(x\right)=$ .

$m'\left(x\right)=3x^2.\ln x-x^2.\frac{1}{x}$

$m'\left(x\right)=3x^2.\ln x+x^2.\frac{1}{x}$

$m'\left(x\right)=3x^2.\ln x+\frac{1}{x}$

If $p\left(x\right)=\sqrt{x}\cdot\cos x$ , find $p'\left(x\right)$ .

$p'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\cos x+\sin x\cdot\sqrt{x}$

$p'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\cos x-\sin x$

$p'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\cos x-\sin x\cdot\sqrt{x}$

Given $r\left(x\right)=\cos x\cdot\ln x$ , determine $r'\left(x\right)$ .

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

$r'\left(x\right)=\sin x\cdot\ln x+\frac{\cos x}{x}$

$r'\left(x\right)=\sin x\cdot\ln x+\cos x$

Determine the derivative of the function $h\left(x\right)=e^x\cdot\sin x$ .

$h'\left(x\right)=e^x\cdot\sin x+e^x\cdot\cos x$ .

$h'\left(x\right)=e^x\cdot\sin x-e^x\cdot\cos x$

$h'\left(x\right)=e^x\cdot\sin x+\cos x$

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The Derivative Function Of The Product Of Two Functions

Calculate the derivative of ﻿s(x)=x⋅ex⋅sin⁡xs\left(x\right)=x\cdot e^x\cdot\sin xs(x)=x⋅ex⋅sinx﻿ .

﻿s′(x)=ex⋅sin⁡x+x⋅ex⋅cos⁡x−x⋅exsin⁡xs'\left(x\right)=e^x\cdot\sin x+x\cdot e^x\cdot\cos x-x\cdot e^x\sin xs′(x)=ex⋅sinx+x⋅ex⋅cosx−x⋅exsinx﻿

﻿s′(x)=ex⋅sin⁡x−x⋅ex⋅cos⁡x−x⋅exsin⁡xs'\left(x\right)=e^x\cdot\sin x-x\cdot e^x\cdot\cos x-x\cdot e^x\sin xs′(x)=ex⋅sinx−x⋅ex⋅cosx−x⋅exsinx﻿

﻿s′(x)=ex⋅sin⁡x+x⋅ex⋅cos⁡x+x⋅exsin⁡xs'\left(x\right)=e^x\cdot\sin x+x\cdot e^x\cdot\cos x+x\cdot e^x\sin xs′(x)=ex⋅sinx+x⋅ex⋅cosx+x⋅exsinx﻿

Consider ﻿w(x)=x⋅cos⁡x⋅ln⁡xw\left(x\right)=x\cdot\cos x\cdot\ln xw(x)=x⋅cosx⋅lnx﻿ . Find ﻿w′(x)w'\left(x\right)w′(x)﻿ .

﻿w′(x)=cos⁡x⋅ln⁡x−x⋅sin⁡x⋅ln⁡x+x⋅cos⁡x⋅1xw'\left(x\right)=\cos x\cdot\ln x-x\cdot\sin x\cdot\ln x+x\cdot\cos x\cdot\frac{1}{x}w′(x)=cosx⋅lnx−x⋅sinx⋅lnx+x⋅cosx⋅x1​﻿

﻿w′(x)=cos⁡x⋅ln⁡x−x⋅sin⁡x⋅ln⁡x−x⋅cos⁡x⋅1xw'\left(x\right)=\cos x\cdot\ln x-x\cdot\sin x\cdot\ln x-x\cdot\cos x\cdot\frac{1}{x}w′(x)=cosx⋅lnx−x⋅sinx⋅lnx−x⋅cosx⋅x1​﻿

﻿w′(x)=cos⁡x⋅ln⁡x+x⋅sin⁡x⋅ln⁡x+x⋅cos⁡x⋅1xw'\left(x\right)=\cos x\cdot\ln x+x\cdot\sin x\cdot\ln x+x\cdot\cos x\cdot\frac{1}{x}w′(x)=cosx⋅lnx+x⋅sinx⋅lnx+x⋅cosx⋅x1​﻿

If ﻿v(x)=sin⁡x⋅exv\left(x\right)=\sin x\cdot e^xv(x)=sinx⋅ex﻿ , calculate ﻿v′(x)v'\left(x\right)v′(x)﻿ .

﻿v′(x)=cos⁡x⋅ex+sin⁡x⋅ev'\left(x\right)=\cos x\cdot e^x+\sin x\cdot e^{ }v′(x)=cosx⋅ex+sinx⋅e﻿

﻿v′(x)=cos⁡x⋅ex−sin⁡x⋅exv'\left(x\right)=\cos x\cdot e^x-\sin x\cdot e^xv′(x)=cosx⋅ex−sinx⋅ex﻿

﻿v′(x)=cos⁡x⋅ex+sin⁡x⋅exv'\left(x\right)=\cos x\cdot e^x+\sin x\cdot e^xv′(x)=cosx⋅ex+sinx⋅ex﻿

Consider ﻿q(x)=1x⋅exq\left(x\right)=\frac{1}{x}\cdot e^xq(x)=x1​⋅ex﻿ , find ﻿q′(x)q'\left(x\right)q′(x)﻿ .

﻿q′(x)=−1x2⋅ex+1x⋅exq'\left(x\right)=-\frac{1}{x^2}\cdot e^x+\frac{1}{x}\cdot e^xq′(x)=−x21​⋅ex+x1​⋅ex﻿

﻿q′(x)=−1x2⋅ex−1x⋅exq'\left(x\right)=-\frac{1}{x^2}\cdot e^x-\frac{1}{x}\cdot e^xq′(x)=−x21​⋅ex−x1​⋅ex﻿

﻿q′(x)=−1x2⋅ex+1x⋅eq'\left(x\right)=-\frac{1}{x^2}\cdot e^x+\frac{1}{x}\cdot e^{ }q′(x)=−x21​⋅ex+x1​⋅e﻿

Find the derivative of ﻿t(x)=ln⁡x⋅e−xt\left(x\right)=\ln x\cdot e^{-x}t(x)=lnx⋅e−x﻿ .

﻿t′(x)=1x⋅e−x−ln⁡x⋅e−xt'\left(x\right)=\frac{1}{x}\cdot e^{-x}-\ln x\cdot e^{-x}t′(x)=x1​⋅e−x−lnx⋅e−x﻿

﻿t′(x)=1x⋅e−x+ln⁡x⋅e−xt'\left(x\right)=\frac{1}{x}\cdot e^{-x}+\ln x\cdot e^{-x}t′(x)=x1​⋅e−x+lnx⋅e−x﻿

﻿t′(x)=1x⋅e−x−ln⁡x⋅ext'\left(x\right)=\frac{1}{x}\cdot e^{-x}-\ln x\cdot e^xt′(x)=x1​⋅e−x−lnx⋅ex﻿

Given ﻿u(x)=x⋅tan⁡xu\left(x\right)=\sqrt{x}\cdot\tan xu(x)=x​⋅tanx﻿ , determine ﻿u′(x)u'\left(x\right)u′(x)﻿ .

﻿u′(x)=12x⋅tan⁡x+x⋅sec⁡2xu'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\tan x+\sqrt{x}\cdot\sec^2xu′(x)=2x​1​⋅tanx+x​⋅sec2x﻿

﻿u′(x)=12x⋅tan⁡x+x⋅sec⁡2xu'\left(x\right)=12\sqrt{x}\cdot\tan x+\sqrt{x}\cdot\sec^2xu′(x)=12x​⋅tanx+x​⋅sec2x﻿

﻿u′(x)=12x⋅tan⁡x−x⋅sec⁡2xu'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\tan x-\sqrt{x}\cdot\sec^2xu′(x)=2x​1​⋅tanx−x​⋅sec2x﻿

Let ﻿m(x)=x3⋅ln⁡xm\left(x\right)=x^3\cdot\ln xm(x)=x3⋅lnx﻿ , calculate ﻿m′(x)=m'\left(x\right)=m′(x)=﻿ .

﻿m′(x)=3x2.ln⁡x−x2.1xm'\left(x\right)=3x^2.\ln x-x^2.\frac{1}{x}m′(x)=3x2.lnx−x2.x1​﻿

﻿m′(x)=3x2.ln⁡x+x2.1xm'\left(x\right)=3x^2.\ln x+x^2.\frac{1}{x}m′(x)=3x2.lnx+x2.x1​﻿

﻿m′(x)=3x2.ln⁡x+1xm'\left(x\right)=3x^2.\ln x+\frac{1}{x}m′(x)=3x2.lnx+x1​﻿

If ﻿p(x)=x⋅cos⁡xp\left(x\right)=\sqrt{x}\cdot\cos xp(x)=x​⋅cosx﻿ , find ﻿p′(x)p'\left(x\right)p′(x)﻿ .

﻿p′(x)=12x⋅cos⁡x+sin⁡x⋅xp'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\cos x+\sin x\cdot\sqrt{x}p′(x)=2x​1​⋅cosx+sinx⋅x​﻿

﻿p′(x)=12x⋅cos⁡x−sin⁡xp'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\cos x-\sin xp′(x)=2x​1​⋅cosx−sinx﻿

﻿p′(x)=12x⋅cos⁡x−sin⁡x⋅xp'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\cos x-\sin x\cdot\sqrt{x}p′(x)=2x​1​⋅cosx−sinx⋅x​﻿

Given ﻿r(x)=cos⁡x⋅ln⁡xr\left(x\right)=\cos x\cdot\ln xr(x)=cosx⋅lnx﻿ , determine ﻿r′(x)r'\left(x\right)r′(x)﻿ .

﻿r′(x)=sin⁡x⋅ln⁡x−cos⁡xxr'\left(x\right)=\sin x\cdot\ln x-\frac{\cos x}{x}r′(x)=sinx⋅lnx−xcosx​﻿

﻿r′(x)=sin⁡x⋅ln⁡x+cos⁡xxr'\left(x\right)=\sin x\cdot\ln x+\frac{\cos x}{x}r′(x)=sinx⋅lnx+xcosx​﻿

﻿r′(x)=sin⁡x⋅ln⁡x+cos⁡xr'\left(x\right)=\sin x\cdot\ln x+\cos xr′(x)=sinx⋅lnx+cosx﻿

Determine the derivative of the function ﻿h(x)=ex⋅sin⁡xh\left(x\right)=e^x\cdot\sin xh(x)=ex⋅sinx﻿ .

﻿h′(x)=ex⋅sin⁡x+ex⋅cos⁡xh'\left(x\right)=e^x\cdot\sin x+e^x\cdot\cos xh′(x)=ex⋅sinx+ex⋅cosx﻿ .

﻿h′(x)=ex⋅sin⁡x−ex⋅cos⁡xh'\left(x\right)=e^x\cdot\sin x-e^x\cdot\cos xh′(x)=ex⋅sinx−ex⋅cosx﻿

﻿h′(x)=ex⋅sin⁡x+cos⁡xh'\left(x\right)=e^x\cdot\sin x+\cos xh′(x)=ex⋅sinx+cosx﻿

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Great!

Calculate the derivative of $s\left(x\right)=x\cdot e^x\cdot\sin x$ .

$s'\left(x\right)=e^x\cdot\sin x+x\cdot e^x\cdot\cos x-x\cdot e^x\sin x$

$s'\left(x\right)=e^x\cdot\sin x-x\cdot e^x\cdot\cos x-x\cdot e^x\sin x$

$s'\left(x\right)=e^x\cdot\sin x+x\cdot e^x\cdot\cos x+x\cdot e^x\sin x$

Consider $w\left(x\right)=x\cdot\cos x\cdot\ln x$ . Find $w'\left(x\right)$ .

$w'\left(x\right)=\cos x\cdot\ln x-x\cdot\sin x\cdot\ln x+x\cdot\cos x\cdot\frac{1}{x}$

$w'\left(x\right)=\cos x\cdot\ln x-x\cdot\sin x\cdot\ln x-x\cdot\cos x\cdot\frac{1}{x}$

$w'\left(x\right)=\cos x\cdot\ln x+x\cdot\sin x\cdot\ln x+x\cdot\cos x\cdot\frac{1}{x}$

If $v\left(x\right)=\sin x\cdot e^x$ , calculate $v'\left(x\right)$ .

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

$v'\left(x\right)=\cos x\cdot e^x-\sin x\cdot e^x$

$v'\left(x\right)=\cos x\cdot e^x+\sin x\cdot e^x$

Consider $q\left(x\right)=\frac{1}{x}\cdot e^x$ , find $q'\left(x\right)$ .

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

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

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

Find the derivative of $t\left(x\right)=\ln x\cdot e^{-x}$ .

$t'\left(x\right)=\frac{1}{x}\cdot e^{-x}-\ln x\cdot e^{-x}$

$t'\left(x\right)=\frac{1}{x}\cdot e^{-x}+\ln x\cdot e^{-x}$

$t'\left(x\right)=\frac{1}{x}\cdot e^{-x}-\ln x\cdot e^x$

Given $u\left(x\right)=\sqrt{x}\cdot\tan x$ , determine $u'\left(x\right)$ .

$u'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\tan x+\sqrt{x}\cdot\sec^2x$

$u'\left(x\right)=12\sqrt{x}\cdot\tan x+\sqrt{x}\cdot\sec^2x$

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

Let $m\left(x\right)=x^3\cdot\ln x$ , calculate $m'\left(x\right)=$ .

$m'\left(x\right)=3x^2.\ln x-x^2.\frac{1}{x}$

$m'\left(x\right)=3x^2.\ln x+x^2.\frac{1}{x}$

$m'\left(x\right)=3x^2.\ln x+\frac{1}{x}$

If $p\left(x\right)=\sqrt{x}\cdot\cos x$ , find $p'\left(x\right)$ .

$p'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\cos x+\sin x\cdot\sqrt{x}$

$p'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\cos x-\sin x$

$p'\left(x\right)=\frac{1}{2\sqrt{x}}\cdot\cos x-\sin x\cdot\sqrt{x}$

Given $r\left(x\right)=\cos x\cdot\ln x$ , determine $r'\left(x\right)$ .

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

$r'\left(x\right)=\sin x\cdot\ln x+\frac{\cos x}{x}$

$r'\left(x\right)=\sin x\cdot\ln x+\cos x$

Determine the derivative of the function $h\left(x\right)=e^x\cdot\sin x$ .

$h'\left(x\right)=e^x\cdot\sin x+e^x\cdot\cos x$ .

$h'\left(x\right)=e^x\cdot\sin x-e^x\cdot\cos x$

$h'\left(x\right)=e^x\cdot\sin x+\cos x$