Daily Math Guide

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## Saturday, March 6, 2021

EXPONENTS, POWERS, RAISED TO THE POWER

The "exponents", sometimes called "powers" or "raised to the power of" is the "teller" of how many base or bases the power has. Those words are of the same in meaning. It depends on how the person comfortable with the use of words. Dealing with this expression requires "base" and of course, exponent or power. This can usually be seen in mathematics such as Algebra, Geometry, Trigonometry, Calculus, etc. Powers play one of the biggest roles in the field of mathematics and sciences. It also tells the readers the meaning of the graphs, say exponential graphs, waves, bacterial growth, and a lot more. We provide a short description below to further our ideas and understanding of exponential.

In;

..the power "2" in x^2 tells how many base "x" has in the expression given. Those mentioned above is the general rule. It is the overall picture of its simplicity.

By the way, below are the labels of the expressions given;

Notice that we have three (3) bases, they are x, y, and n with corresponding powers 2,3,and m respectively.

Now we had a basic knowledge about its components. With this, we can easily dissect its parts for a better approach. Illustrative examples were provided to guide how its details look like. Then, the "check for yourself" was also provided to train you slowly until finally, you can make it through "Simple tests". Remember that you don't need to do this in a rush manner. It should be slowly but focused.

Illustrative Examples:

Express the given exponentials.

# Check It For Yourself!

Simple Tests!

## DERIVATIVE OF TRIGONOMETRIC FUNCTIONS

### Formulae of "The derivatives of Trigonometric Functions "

$i.)\frac{d}{dx}sin{\color{DarkRed} u}=cos{\color{DarkRed} u}\frac{du}{dx}$

$ii.)\frac{d}{dx}cos{\color{DarkRed} u}=-sin{\color{DarkRed} u}\frac{du}{dx}$

$iii.)\frac{d}{dx}tan{\color{DarkRed} u}=sec^2{\color{DarkRed} u}\frac{du}{dx}$

$iv.)\frac{d}{dx}cot{\color{DarkRed} u}=-csc^2{\color{DarkRed} u}\frac{du}{dx}$

$v.)\frac{d}{dx}sec{\color{DarkRed} u}=sec{\color{DarkRed} u}tan{\color{DarkRed} u}\frac{du}{dx}$

$vi.)\frac{d}{dx}csc{\color{DarkRed} u}=-csc{\color{DarkRed} u}cot{\color{DarkRed} u}\frac{du}{dx}$

#### Illustrative Examples

Find the first derivative of the given trigonometric function.

$1.)y=sin5x$

In the given example y = sin 5x, u = 5x. We can use the relationship "i" provided above.

$\frac{dy}{dx}=\frac{d}{dx}(sin5x)$

Above show the first approach as we derive the u.

$\frac{dy}{dx}=cos5x\frac{d}{dx}(5x)$

Having the relationship derivative of sinu = cosu.

$\frac{dy}{dx}=5cos5x$

$2.)u=cos7v$

$\frac{du}{dv}=\frac{d}{dv}cos7v$

$\frac{du}{dv}=-sin7v*\frac{d}{dv}(7v)$

$\frac{du}{dv}=-sin7v.(7)$

$\frac{du}{dv}=-7sin7v$

$3.)w=tan4a$

$\frac{dw}{da}=sec^22a.\frac{d}{da}(2a)$

$\frac{dw}{da}=sec^22a.(2)$

$\frac{dw}{da}=2sec^22a$

$4.) x=cot3\theta$

$\frac{ dx}{d\theta }=\frac{d}{d\theta }cot3\theta$

$\frac{ dx}{d\theta }=-csc^23\theta .\frac{d}{d\theta }(3\theta )$

$\frac{ dx}{d\theta }=-csc^23\theta .(3)$

$\frac{ dx}{d\theta }=-3csc^23\theta$

$5.)y=sec2x$

$\frac{dy}{dx}=\frac{d}{dx}(sec2x)$

$\frac{dy}{dx}=sec2xtan2x.\frac{d}{dx}(2x)$

$\frac{dy}{dx}=sec2xtan2x.(2)$

$\frac{dy}{dx}=2sec2xtan2x$