Math 181 Miniproject 3: Texting Lesson.md
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My lesson Topic
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<div><img class="left"/><div class="alert gray">
Jon could you help me? I just can't understand this problem. It goes like this
Find the derivative of:

$$f(x)=5x^3+
√4 +7+6^x$$
,
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<div><div class="alert blue">

Sure Garfield ! First identify what you are working with here. Remember the derivative rules. A crucial fact to keep in mind is that these rules cannot be applied to all functions so make sure you know the rules you will be applying

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<div><img class="left"/><div class="alert gray ">

Derivative rules? not sure what that is all about. Care to explain?

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<div><div class="alert blue">
Well Garfield, there are several derivative rules but let's focus on the ones your problem covers. Based on my observations you will apply the sum of derivatives rule, The constant function rule, power rule ,and the rule of exponential functions

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<div><div class="alert blue">
The rules are written this way

**Sum of derivatives rule:**
$d/dx f(x)+g(x)=f'(x)+g'(x)$

**power function rule**
f(x)=$x^n f'(x)=nx$^n-1^
(for any non zero value)

**constant function rule**
for any real number c -> f'( c )=0

**rule of exponential functions**
$f(x)=a^x f'(x)=a^x*ln(a)$
(for any real positive number)

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<div><img class="left"/><div class="alert gray">

How exactly does everything come into play?

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<div><div class="alert blue">

We can find this step by step, cutting everything into a smaller function. I will give you an example.
In your problem the first thing that is shown is $5x^3$
f(x)=$5^3$
Applying the power rule to this will mean..

f'(x)=3*5x^3-1^
f'(x)=15x^2^
Now that you have that down proceed to
√4 knowing it is simplified to 2, a constant

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<div><img class="left"/><div class="alert gray">

Okay 2 is constant and so is 7, both fall under the constant rule meaning derivatives are zero. But what about $6^x$ ?

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<div><div class="alert blue">
We will simply apply the rule of exponential functions

$f(x)=a^x f'(x)=a^x*ln(a)$
where it will be rewritten like this:
$f(x)=6^x$
$f'(x)=6^x*ln(6)$

At this point you have everything you need and all you do is rewrite the entire function applying the sum of derivatives rule, which as the rule writes you simply add the derivatives
$d/dx f(x)+g(x)=f'(x)+g'(x)$

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<div><img class="left"/><div class="alert gray">

so $f'(x)=15x^2 +6^x ln(6)?$

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<div><div class="alert blue">
That is exactly it!
The derivative of..

$f(x)=5x^3+
√4 +7+6^x$

is indeed...
$f'(x)=15x^2 +6^x ln(6)$

it is important to know these rules to make sure you are calculating the correct derivative values. It can be real simple as you cn see if you do your work in chunks and then rewrite it as one in the end, at least for your type of problem.
Hoped this helped in some way

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<div><img class="left"/><div class="alert gray">
I was sort of confused at first but I know what I must remember now. The derivative rules are straightforward and are real specific so I know when to use them. Thanks!