Find d p d q i f p = 1 q + 9 \frac{dp}{dq}\ if\ p=\frac{1}{\sqrt{q+9}} d q d p i f p = q + 9 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> 1

− 1 2 ( q + 9 ) 3 2 -\frac{1}{2\left(q+9\right)^{\frac{3}{2}}} − 2 ( q + 9 ) 2 3 1

1 2 ( q + 9 ) 3 2 \frac{1}{2\left(q+9\right)^{\frac{3}{2}}} 2 ( q + 9 ) 2 3 1

− 1 ( q + 9 ) 3 2 -\frac{1}{\left(q+9\right)^{\frac{3}{2}}} − ( q + 9 ) 2 3 1

− 1 q + 9 -\frac{1}{\sqrt{q+9}} − q + 9 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> 1

d y d x i f y = e x ( sin ⁡ x − cos ⁡ x ) \frac{dy}{dx}\ if\ y=e^x\left(\sin x-\cos x\right) d x d y i f y = e x ( sin x − cos x ) Find

2 e x sin ⁡ x 2e^x\sin x 2 e x sin x

2 e x ( sin ⁡ x − cos ⁡ x ) + 2 e x 2e^x\left(\sin x-\cos x\right)+2e^x 2 e x ( sin x − cos x ) + 2 e x

4 e x ( sin ⁡ x − cos ⁡ x ) 4e^x\left(\sin x-\cos x\right) 4 e x ( sin x − cos x )

d y d t i f y = cos ⁡ 10 t + 12 \frac{dy}{dt}\ if\ y=\cos\sqrt{10t+12} d t d y i f y = cos 1 0 t + 1 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> Find

− 5 10 t + 12 sin ⁡ 10 t + 12 \frac{-5}{\sqrt{10t+12}}\sin\sqrt{10t+12} 1 0 t + 1 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 5 sin 1 0 t + 1 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

− 1 2 10 t + 12 sin ⁡ 10 t + 12 \frac{-1}{2\sqrt{10t+12}}\sin\sqrt{10t+12} 2 1 0 t + 1 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 1 sin 1 0 t + 1 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

− sin ⁡ 10 t + 12 -\sin\sqrt{10t+12} − sin 1 0 t + 1 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

− sin ⁡ ( 5 10 t + 12 ) -\sin\left(\frac{5}{\sqrt{10t+12}}\right) − sin ( 1 0 t + 1 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> 5 )

d y d x i f y = ln ⁡ 2 x \frac{dy}{dx}if\ y=\ln2x d x d y i f y = ln 2 x Find

− 1 2 x \frac{-1}{2x} 2 x − 1

d y d x i f y = ln ⁡ ( ln ⁡ 2 x ) \frac{dy}{dx}\ if\ y=\ln\left(\ln2x\right) d x d y i f y = ln ( ln 2 x ) Find

1 x ln ⁡ 2 x \frac{1}{x\ln2x} x ln 2 x 1

1 ln ⁡ 2 x \frac{1}{\ln2x} ln 2 x 1

d y d x i f y = e ( 7 − 9 x ) \frac{dy}{dx}\ if\ y=e^{\left(7-9x\right)} d x d y i f y = e ( 7 − 9 x ) Find

− 9 e ( 7 − 9 x ) -9e^{\left(7-9x\right)} − 9 e ( 7 − 9 x )

− 9 ln ⁡ ( 7 − 9 x ) -9\ln\left(7-9x\right) − 9 ln ( 7 − 9 x )

7 e ( 7 − 9 x ) 7e^{\left(7-9x\right)} 7 e ( 7 − 9 x )

Chain Rule

Authored by Danielle Meyer

Mathematics

11th - 12th Grade

CCSS covered

Used 310+ times

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12 questions

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MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Find $\frac{dv}{dt}\ if\ v=t+\frac{8}{t}$

$1-\frac{8}{t^2}$

$1-\frac{8}{t}$

$t-\frac{8}{t^2}$

$1+\frac{8}{t^2}$

Tags

CCSS.HSA.APR.D.6

CCSS.HSA.SSE.B.3

CCSS.HSA.APR.D.7

CCSS.HSA.SSE.A.2

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Find $\frac{dp}{dq}\ if\ p=\frac{1}{\sqrt{q+9}}$

$-\frac{1}{2\left(q+9\right)^{\frac{3}{2}}}$

$\frac{1}{2\left(q+9\right)^{\frac{3}{2}}}$

$-\frac{1}{\left(q+9\right)^{\frac{3}{2}}}$

$-\frac{1}{\sqrt{q+9}}$

Tags

CCSS.HSN.RN.A.2

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Find $\frac{dy}{dx}\ if\ y=\frac{x^3}{x-1}$

$\frac{2x^3-3x^2}{\left(x-1\right)^2}$

$\frac{2x^3+3x^2}{\left(x-1\right)^2}$

$\frac{-2x^3-3x^2}{\left(x-1\right)^2}$

$\frac{-2x^3+3x^2}{\left(x-1\right)^2}$

Tags

CCSS.HSA.APR.D.6

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

$\frac{ds}{dt}\ if\ s=\frac{3e^t}{2e^t+1}$ Find

$\frac{3e^t}{\left(2e^t+1\right)^3}$

$\frac{3e^t}{\left(2e^t+1\right)^2}$

$\frac{3e^t}{\left(2e^t+1\right)^{ }}$

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

Tags

CCSS.HSF.IF.A.1

CCSS.HSF.IF.A.2

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

$\frac{dy}{dx}\ if\ y=^3\sqrt{x^2\ }+x^e$ Find

$\frac{2x^{\frac{1}{3}}}{3}+ex^e$

$\frac{2}{3x^{\frac{1}{3}}}+ex^{\left(e-1\right)}$

$\frac{2x^{\frac{1}{3}}}{3}+ex^{\left(e-1\right)}$

$\frac{2}{3x^{\frac{1}{3}}}+ex^e$

Tags

CCSS.HSN.RN.A.2

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

$\frac{dy}{dx}\ if\ y=\left(2x^2+4\right)^3$ Find

$\left(12x+4\right)\left(2x^2+4\right)^2$

$3\left(2x^2+4\right)^2$

$12x\left(2x^2+4\right)^2$

$12\left(2x^2+4\right)^2$

Tags

CCSS.HSA.APR.A.1

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

$\frac{dr}{dt}\ if\ r=\left(\sec\theta+\tan\theta\right)^{-3}$ Find

$\frac{-3\sec\theta}{\left(\sec\theta+\tan\theta\right)^3}$

$-3\left(\sec\theta+\tan\theta\right)^{-4}\left(\tan^2\theta+\sec\theta\tan\theta\right)$

$-3\left(\sec\theta+\tan\theta\right)^{-4}$

$-3\left(\sec\theta+\tan\theta+\sec^2\theta\right)^{-4}$

Tags

CCSS.HSA.SSE.B.3

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

$\frac{dy}{dx}\ if\ y=e^x\left(\sin x-\cos x\right)$ Find

$2e^x\sin x$

$2e^x\left(\sin x-\cos x\right)+2e^x$

$0$

$4e^x\left(\sin x-\cos x\right)$

Tags

CCSS.HSF.IF.A.1

CCSS.HSF.IF.A.2

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

$\frac{dy}{dt}\ if\ y=\cos\sqrt{10t+12}$ Find

$\frac{-5}{\sqrt{10t+12}}\sin\sqrt{10t+12}$

$\frac{-1}{2\sqrt{10t+12}}\sin\sqrt{10t+12}$

$-\sin\sqrt{10t+12}$

$-\sin\left(\frac{5}{\sqrt{10t+12}}\right)$

10.

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

$\frac{dy}{dx}if\ y=\ln2x$ Find

$-\frac{1}{x}$

$\frac{1}{2x}$

$\frac{1}{x}$

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

Tags

CCSS.HSF.IF.A.1

CCSS.HSF.IF.A.2

CCSS.HSF.BF.B.5

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Chain Rule

Find dvdt if v=t+8t\frac{dv}{dt}\ if\ v=t+\frac{8}{t}dtdv​ if v=t+t8​

Find dpdq if p=1q+9\frac{dp}{dq}\ if\ p=\frac{1}{\sqrt{q+9}}dqdp​ if p=q+9​1​

Find dydx if y=x3x−1\frac{dy}{dx}\ if\ y=\frac{x^3}{x-1}dxdy​ if y=x−1x3​

dsdt if s=3et2et+1\frac{ds}{dt}\ if\ s=\frac{3e^t}{2e^t+1}dtds​ if s=2et+13et​ Find

dydx if y=3x2 +xe\frac{dy}{dx}\ if\ y=^3\sqrt{x^2\ }+x^edxdy​ if y=3x2 ​+xe Find

dydx if y=(2x2+4)3\frac{dy}{dx}\ if\ y=\left(2x^2+4\right)^3dxdy​ if y=(2x2+4)3 Find

drdt if r=(sec⁡θ+tan⁡θ)−3\frac{dr}{dt}\ if\ r=\left(\sec\theta+\tan\theta\right)^{-3}dtdr​ if r=(secθ+tanθ)−3 Find

dydx if y=ex(sin⁡x−cos⁡x)\frac{dy}{dx}\ if\ y=e^x\left(\sin x-\cos x\right)dxdy​ if y=ex(sinx−cosx) Find

dydt if y=cos⁡10t+12\frac{dy}{dt}\ if\ y=\cos\sqrt{10t+12}dtdy​ if y=cos10t+12​ Find

dydxif y=ln⁡2x\frac{dy}{dx}if\ y=\ln2xdxdy​if y=ln2x Find

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Discover more resources for Mathematics

Find $\frac{dv}{dt}\ if\ v=t+\frac{8}{t}$

Find $\frac{dp}{dq}\ if\ p=\frac{1}{\sqrt{q+9}}$

Find $\frac{dy}{dx}\ if\ y=\frac{x^3}{x-1}$

$\frac{ds}{dt}\ if\ s=\frac{3e^t}{2e^t+1}$ Find

$\frac{dy}{dx}\ if\ y=^3\sqrt{x^2\ }+x^e$ Find

$\frac{dy}{dx}\ if\ y=\left(2x^2+4\right)^3$ Find

$\frac{dr}{dt}\ if\ r=\left(\sec\theta+\tan\theta\right)^{-3}$ Find

$\frac{dy}{dx}\ if\ y=e^x\left(\sin x-\cos x\right)$ Find

$\frac{dy}{dt}\ if\ y=\cos\sqrt{10t+12}$ Find

$\frac{dy}{dx}if\ y=\ln2x$ Find