## The Differentials: Example 7

### Find $d\left[ln{\left(y^2\right)}+ln{\left(1+\sqrt y\right)}\right]$.

SOLUTION:

$=d\left[ln{\left(y^2\left(1+\sqrt y\right)\right)}\right]$

$=d\left[ln{\left(y^2+y^\frac{5}{2}\right)}\right]$

$=\frac{1}{y^2+y^\frac{5}{2}}\times\left(2y+\frac{5}{2}y^\frac{3}{2}\right)dy$

$=\frac{y\left(2+\frac{5}{2}y^\frac{1}{2}\right)}{y^2\left(1+\sqrt y\right)}dy$

$=\frac{2+\frac{5}{2}y^\frac{1}{2}}{y\left(1+\sqrt y\right)}dy$

$=\frac{2+\frac{5}{2}\sqrt y}{y\left(1+\sqrt y\right)}dy$

## The Differentials: Example 6

### Find $d\left({cos}^3{\left(t\right)}\right)$.

SOLUTION:

$=-3{cos}^2{t}sin{t}dt$

## The Differentials: Example 5

### Find $d\left(3w^{-1}+\frac{w^2}{4}-7\right)$.

SOLUTION:

$=\left(-\frac{3}{w^2}\right)dw+\left(\frac{w}{2}\right)dw-0$

$=\left(\frac{-3}{w^2}+\frac{w}{2}\right)dw$

## The Differentials: Example 4

### Find $d\left(cos\:\theta \:sin\:\theta \right)$.

SOLUTION:

We use the product rule to solve this one. So, we have

$=cos{\theta}\ d\left(sin{\theta}\right)+sin{\theta}d\left(cos{\theta}\right)$

$=cos{\theta}cos{\theta}d\theta+sin{\theta}\left(-sin{\theta}\right)d\theta$

$=\left({cos}^2{\theta}-{sin}^2{\theta}\right)d\theta$

$=cos{\left(2\theta\right)}d\theta$

## The Differentials: Example 3

### Find $d\left(\frac{1}{\sqrt{x^3+7}}+4x^2-1\right)$.

Solution:

$=d\left(\frac{1}{\sqrt{x^3+7}}\right)+d\left(4x^2\right)+d\left(-1\right)$

$=\left(-\frac{3x^2}{2\left(\sqrt{x^3+7}\right)^3}+8x\right)dx$

## The Differentials: Example 2

### Find $dy$ if $y=\frac{2x}{3x-1}$.

Solution:

We get the differential of both sides of the equation.

$dy=d\left(\frac{2x}{3x-1}\right)$

We apply the different differential formulas

$dy=\frac{\left(3x-1\right)\:d\left(2x\right)-2x\:d\left(3x-1\right)}{\left(3x-1\right)^2}$

$dy=\frac{\left(3x-1\right)\:\left(2\:dx\right)-2x\:\left(3\:dx\right)}{\left(3x-1\right)^2}$

$dy=\frac{\left(6x-2\right)dx-6x\left(dx\right)}{\left(3x-1\right)^2}$

$dy=\frac{\left(6x-2-6x\right)dx}{\left(3x-1\right)^2}$

$dy=\frac{-2\:dx}{\left(3x-1\right)^2}$

## The Differentials: Example 1

### Find $dy$ if $y=x^3-4x^2+5x$.

Solution:

We get the differential of both sides of the equation.

$dy=d\left(x^3-4x^2+5x\right)$

We apply the different differential formulas

$dy=d\left(x^3\right)+d\left(-4x^2\right)+d\left(5x\right)$

$dy=3x^2\:dx-8x\:dx+5\:dx$

$dy=\left(3x^2-8x+5\right)dx$