## 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

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

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

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

## The Differential

On your Calculus I, the notation $\frac{dy}{dx}$ was regarded as a single symbol to denote the derivative of a function y=f(x) with respect to x. More specifically, you have used the symbol $\frac{dy}{dx}$ to denote the limit of the quotient $\frac{\Delta y}{\Delta x}$ as $\Delta x$ approaches 0. This module introduces the concept of differentials by giving two separate meanings of both dy and dx. We now shall treat them as separate variables.

The symbol dx means differential of x, which is the independent variable in the function y=f(x). The differential of the independent variable is equal to the increment of the variable. Mathematically, we write it as

$dx=\Delta \:x$

On a similar note, dy means the differential of y, which is the dependent variable in the function y=f(x). The differential of a function is equal to its derivative multiplied by the differential of its independent variable. Mathematically, we denote it as

$dy=f'\left(x\right)dx$

Notice that when we divide dx to both sides of the equation, we will come up with

$\frac{dy}{dx}=f'\left(x\right)$

which brings us back the derivative symbol. This means that we can actually use differentials to solve for the derivative of functions.

We can use the definition of a differential to rewrite each of the derivative rules in differential form. The following summary lists the differential forms corresponding to several differentiation rules

Let u and v be differential functions of x, and c be any constant.

The examples that follow illustrate the use of these formulas to solve for differentials and derivatives of functions.

Example 1

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

See the example 1 solution.

In practice, we simply get the derivative of the right member of the equation and multiply it by dx.

Example 2

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

See the example 2 solution.

Example 3

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

See the example 3 solution.

Example 4

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

See the example 4 solution.

Example 5

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

See the example 5 solution.

Example 6

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

See the example 6 solution.

Example 7

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

See the example 7 solution.

## Limit of a Function

### Definition of a Limit

Let f(x) be any function and let a and L be numbers. If we can make f(x) as close to L as we please by choosing x sufficiently close to a, then we say that the limit of f(x) as x approaches a is L or symbolically,

$\lim\limits_{x\to a}\left(f\left(x\right)\right)=L$

### Theorems on Limits

1. $\lim\limits_{x\to a}\left(c\right)=c$     ,   c=any constant
2. $\lim\limits_{x\to a}x=a$,      a=any real number
3. $\lim\limits_{x\to a}c\cdot f\left(x\right)=c\cdot \lim\limits_{x\to a}f\left(x\right)$
4. $\lim\limits_{x\to a}\left[f\left(x\right)+g\left(x\right)\right]=\lim\limits_{x\to a}f\left(x\right)+\lim\limits_{x\to \:a}g\left(x\right)$
5. $\lim\limits_{x\to a}\left[f\left(x\right)\cdot g\left(x\right)\right]=\lim\limits_{x\to a}f\left(x\right)\cdot \lim\limits_{x\to \:a}g\left(x\right)$
6. $\lim\limits_{x\to a}\:\frac{f\left(x\right)}{g\left(x\right)}=\frac{\lim\limits_{x\to a}f\left(x\right)}{\lim\limits_{x\to \:a}g\left(x\right)}$
7. $\lim\limits_{x\to a}\:\sqrt[n]{f\left(x\right)}=\sqrt[n]{\lim\limits_{x\to a}f\left(x\right)}$     , n=any positive integer and f(x)>0 if n is even.
8. $\lim\limits_{x\to a}\left[f\left(x\right)\right]^n=\left[\lim\limits_{x\to \:a}f\left(x\right)\right]^n$

SOURCE: Differential and Integral Calculus by Feliciano and Uy