### Home > CALC > Chapter 10 > Lesson 10.1.5 > Problem10-46

10-46.

Find $\frac { d y } { d x }$ for the following equations.

1. $6 x y - \operatorname { cos } ^ { 2 } x = \sqrt { 2 y }$

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

1. $y ^ { \prime \prime } = \frac { 2 } { x ^ { 2 } - 3 x }$

Since $y^{\prime\prime}$ is given, integrate.

Use partial fraction decomposition:
$\frac{2}{x^2-3x}=\frac{a}{x}+\frac{b}{x-3}$

1. $y = \int _ { 2 } ^ { x } 5 t ^ { 3 } d t$

This will be the derivative of an integral, so apply the Fundamental Theorem of Calculus.

1. $y = \operatorname{ln}(\operatorname{ln}(\operatorname{ln} x)))$

Apply the Chain Rule twice.