CPM Homework Help : CALC Problem 11-122

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11-122.

Find the derivative, $\frac { d y } { d x }$ , for each curve below.

$x\operatorname{sin} y − 10y^2 = y\operatorname{ln} x$
Step (a):
$\sin(y)+x\cos(y)y^\prime-20yy^\prime=y^\prime\ln(x)+\frac{y}{x}$

$r = 2 −\operatorname{cos} θ$
Hint (b):
$\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$
More Help (b):
$x=r\cos(\theta)=(2-\cos(\theta))\cos(\theta)$

$\left\{ \begin{array} { l } { x ( t ) = 4 t - t ^ { 2 } } \\ { y ( t ) = \operatorname { cos } 2 t } \end{array} \right.$
Hint (c):
$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

$y = x \operatorname { sec } ( \sqrt { 2 x ^ { 3 } - 4 x } )$
Step 1(d):
$\text{Let }w=\sqrt{2x^3-4x}.$
Step 2(d):
$y^\prime=\sec(w)+x\sec(w)\tan(w)w^\prime$
Step 3(d):
What is $w^\prime$ ? Substitute $w$ and $w^\prime$ into the equation in Step 2.

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