https://youtu.be/tYDDpKzP-VU

In this video I derive how to do the derivative for a function that has another function in it, for example y = sin(x^5+x^3). This rule is the chain rule and is one of the most important and useful derivative rules in all of calculus and mathematics.

Download video notes: https://1drv.ms/b/s!As32ynv0LoaIidgzupbHXCdgeJAxRg?e=EmTrtX

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# Derivative Rules: Proof of Chain Rule

![Proof of Chain Rule Resized AI.jpg](https://files.peakd.com/file/peakd-hive/mes/23y96R1A6sDm5NoLxmCro8EHMxPvofpncA1XH9tDjCWjq55QXmxEQJAro8fBJoZwod2P5.jpg)

## Extremely Important!

## Function Within a Function

Consider the following example of a function u(x) inside another function f(x).

![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHb6kpi5ebocwocvHwYur2KTcHyba9tDUVVY81a5Mj8Er5h4TZz43xmCG33E53K4KZH.png)

We want to derive an expression for the derivative f'(x) in terms of f'(u) and u' because we know these.

# Chain Rule Proof

Before we derive the proof, let's write out the following:

![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbr9ym4evc5qu8z3zsod3Wt7XYY4hDuVbNfisB8uGFTCPMXzg6iNuuZYX4eMea528i.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbKgySRsZsP6iHBFDnHdqtjP3uAyWh35CSDWXfnBUzGB78CcgZphCrhx3pNzm28BQq.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23sxon1fgCi4VjTTEt5SWMoFQydq6zxN1V3waGYrxtDCemTZKfTLkJHFDds78EdQN9Cn3.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8D6MYfatDuxsHvgsL6oDDnm8vgUsEkFNgFa17Xm6QCgRqc7913BVWb739h4Lv51i8z.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8CfKPvucy6bT2sNafvhT1GihRFBovXFoN4YPANaASGN25zdnBjMMNFs7JSm1Xjmb6B.png)

## Example 1

Find the derivative of y = (x<sup>2</sup>+1)<sup>100</sup>.

**Solution:**

![image.png](https://files.peakd.com/file/peakd-hive/mes/Eo2BTut2nd6kZBpHQLvw8aiybs5QngS1LgpD9i11Nch3UdxDeM5JaEk6BYbmKsgNiYc.png)

We basically have to work from outside to inside when doing the chain rule.

## Example 2

Find the derivative of y = sin(x<sup>5</sup> + x<sup>3</sup>)

**Solution:**

Working from outside to inside, we can readily determine the derivative:

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8D6MYfFvsPBZqDZB2jafGTmBCcXcNPQiG4NrDsrLSPEYwuCmsddB6mr7cs8XTm3NXe.png)