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Home > APCALC > Chapter 10 > Lesson 10.1.5 > Problem 10-60

10-60.

What is $\frac { d y } { d x }$ for each of the following equations?

$6xy - \cos^2(x) = \sqrt { 2 y }$
Step (a):
Differentiate:
$6y+6xy^\prime-2\cos(x)(-\sin(x))=\frac{1}{2}(2y)^{-1/2}(2y^\prime)$

$y^{\prime\prime} =\frac { 2 } { x ^ { 2 } - 3 x }$
Hint (b):
Since $y^{\prime\prime}$ is given, integrate.
More Help (b):
Use partial fraction decomposition:
$\frac{2}{x^2-3x}=\frac{a}{x}+\frac{b}{x-3}$

$y = \int _ { 2 } ^ { x } 5 t ^ { 3 } d t$
Hint (c):
This will be the derivative of an integral, so apply the Fundamental Theorem of Calculus.

$y = \ln(\ln(\ln x))$
Hint (d):
Apply the Chain Rule twice.

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