Derivative of Trigonometric Functions - Daily Math Guide

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## 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.

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

Arrived at the final answer.

$2.)u=cos7v$

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

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

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

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

$3.)w=tan4a$

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

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

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

$4.) x=cot3\theta$

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

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

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

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

$5.)y=sec2x$

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

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

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

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

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