https://youtu.be/5e6MisvvMPE

In this video I derive the derivative of logarithmic and natural logarithmic functions using implicit differentiation and the chain rule. The derivative of natural log is much simpler than other logarithmic functions and thus why it is used more in mathematics.

Download video notes: https://1drv.ms/f/s!As32ynv0LoaIic5Emsdibu6TuSQyAA?e=RhHiPz

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# Derivatives of Logarithmic Functions: y = log<sub>a</sub>(x) and y = ln(x)

![Derivative of Log and ln Resized AI.jpg](https://files.peakd.com/file/peakd-hive/mes/23vspGG4DbpUzJfG1sDkJZb84i351oz1BNkGzwauQrXrsH3yvtfxN2cKmRHU5vqQuNbzP.jpg)

# Derivation Using Implicit Differentiation

![image.png](https://files.peakd.com/file/peakd-hive/mes/23sxr5U7kFAeGgHMGshsGaTxXtVrxRtWvgvziRyRbDxFt6DcHGYaeQEPHC1LLDYPxvbn8.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23tT24arHpky1DBCXCm38JNErxQtVix7Y1zWt8vZUhAsMfWzfks3SJnfjEAFNz9wFaosW.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23sxpMFs9tMEJDMph9PapTQVpXNToPRv67XTyX6dJ49kWnRXMgxko456VFgJ59iR5JRWW.png)

## Example 1

y = ln(x<sup>3</sup> + 1)

**Solution:**

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8DBnQFDCAyoXSq2C8xYQwrLWDqWvypms8VyDiGNUjaYBNjqAM4GdMfVgLu632gzx43.png)

## Example 2

y = ln(sin x)

**Solution:**

![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbr9yko3Ry6cFZ4N4vC19bCVPWCchFQW4yxfJ2RvQqTXpRoEfJbXK3bijJK9dzDkiY.png)