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๐ขDerivatives

2 min readโขnovember 22, 2021

Haseung J

Derivatives are already difficult to tackle for first-timers, but can you imagine __trigonometric functions__** **having derivatives, too?! Well, brace yourselves, because trig functions *do *have derivatives (and their* inverse* functions also have derivatives as well). We'll be in this together! ๐

The thing is: these derivatives are actually pretty easy! Once you memorize them, you wonโt need to give it a second thought. ๐

Before we even look at why the derivatives are the derivatives, letโs just go on and spoil the ending. __If you donโt like spoilers, scroll pass this section__! ๐

So, these are the derivatives! Did it catch you to surprise that the derivative of *sinx* is just *cosx*? And that the derivative of *cosx* is just *-sinx*? Keep scrolling to see the reason behind all that! ๐

**๐ Visuallyย **

The loose graph definition (which is applicable here) of derivatives is the following:ย

f'(x)>0 โก๏ธ f(x) is increasing

f'(x)<0 โก๏ธ f(x) is decreasing

f(x)=0 โก๏ธ extremum

This helps you sketch the derivative if you have* f(x)*, or vice versa. First things first! Let's look at the graph of:

1๏ธโฃ *sinx*.

Graph Courtesy of Desmos

Whenever *f(x)* is **increasing**, the derivative should be **positive**, and whenever *f(x)* is **decreasing**, the derivative should be **negative**. When *f(x)* reaches an extremum, the derivative should cross the x-axis. ๐

If we put that in mind and try sketching the derivative, you should get this:

Graph Courtesy of Desmos

You'll notice that the blue line is basically the graph of cosx! So, this is the visual โproofโ as to why the derivatives are the derivatives. ๐จ

Here are the other two (**red is f(x)** and** blue is the derivative**!):ย

2๏ธโฃ *cosx*:

Graph Courtesy of Desmos

3๏ธโฃ *tanx*:

Graph Courtesy of Desmos

**โ Algebraicallyย **

The derivatives for __cosx__* *and __tanx__ are also very similar, so if you want to take a look at them, check out the linked resources! ๐ก

Of course, nothing is perfected without practice! ๐

Here are some practice problems that you can try out yourself and master by memorizing the derivatives of the trig functions! Always remember that itโs also better to learn them instead of looking them up every time you do homework! ๐ช

Scroll down for the answers!

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Keep scrolling!

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Now you know how the derivatives of trig functions work, you can now ace your homework! ๐ฏ

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