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Revision as of 22:58, 23 October 2008 edit152.1.222.41 (talk) Multiple u's: new section← Previous edit Revision as of 23:02, 23 October 2008 edit undo74.183.242.181 (talk)No edit summaryNext edit →
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<math>\text{Let }v=5x\,</math><br /><br /> <math>\text{Let }v=5x\,</math><br /><br />
<math>\frac{dy}{dx}=\frac{dy}{du}*\frac{du}{dv}*\frac{dv}{dx}</math><br /><br /> <math>\frac{dy}{dx}=\frac{dy}{du}*\frac{du}{dv}*\frac{dv}{dx}</math><br /><br />

Find <math>\frac{dy}{dx}\,</math> then find <math>\frac{d^2y}{dx^2}\,</math> <br /><br />

<math>x^{2} + y^{2} = 1\,</math><br /><br />

Revision as of 23:02, 23 October 2008

x = tan ( y ) {\displaystyle x=\tan \left(y\right)}

1 = sec 2 ( y ) d y d x {\displaystyle 1=\sec ^{2}\left(y\right)*{\frac {dy}{dx}}} (Chain rule, derivative of tan=sec^2)

1 sec 2 ( y ) = d y d x {\displaystyle {\frac {1}{\sec ^{2}\left(y\right)}}={\frac {dy}{dx}}}

cos 2 ( y ) = d y d x {\displaystyle \cos ^{2}\left(y\right)={\frac {dy}{dx}}}

d y d x = cos 2 ( y ) {\displaystyle {\frac {dy}{dx}}=\cos ^{2}\left(y\right)}

9~

x 2 y + x y 2 = 6 {\displaystyle x^{2}y+xy^{2}=6\,}

( 2 x y + x 2 d y d x ) + ( 1 y 2 + x 2 y d y d x ) = 0 {\displaystyle \left(2x*y+x^{2}*{\frac {dy}{dx}}\right)+\left(1*y^{2}+x*2y{\frac {dy}{dx}}\right)=0}

2 x y + x 2 d y d x + y 2 + 2 x y d y d x = 0 {\displaystyle 2xy+x^{2}{\frac {dy}{dx}}+y^{2}+2xy{\frac {dy}{dx}}=0}

x 2 d y d x + 2 x y d y d x = 2 x y y 2 {\displaystyle x^{2}{\frac {dy}{dx}}+2xy{\frac {dy}{dx}}=-2xy-y^{2}}

d y d x = 2 x y y 2 x 2 + 2 x y {\displaystyle {\frac {dy}{dx}}={\frac {-2xy-y^{2}}{x^{2}+2xy}}}

d y d x = 2 x y + y 2 x 2 + 2 x y {\displaystyle {\frac {dy}{dx}}=-{\frac {2xy+y^{2}}{x^{2}+2xy}}}

Multiple u's

To Find dy/dx for
y = 2 cos ( ( 5 x ) 2 ) {\displaystyle y=2\cos \left(\left(5x\right)^{2}\right)}

The way she explains it

you'll make 3 u's
Let  u = 2 cos ( u ) {\displaystyle {\text{Let }}u=2\cos \left(u\right)}

Let  u = u 2 {\displaystyle {\text{Let }}u=u^{2}\,}

Let  u = 5 x {\displaystyle {\text{Let }}u=5x\,}

The way you should do it

Let  y = 2 cos ( u ) {\displaystyle {\text{Let }}y=2\cos \left(u\right)\,}

Let  u = v 2 {\displaystyle {\text{Let }}u=v^{2}\,}

Let  v = 5 x {\displaystyle {\text{Let }}v=5x\,}

d y d x = d y d u d u d v d v d x {\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}*{\frac {du}{dv}}*{\frac {dv}{dx}}}

Find d y d x {\displaystyle {\frac {dy}{dx}}\,} then find d 2 y d x 2 {\displaystyle {\frac {d^{2}y}{dx^{2}}}\,}

x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1\,}